Upload
vuonganh
View
215
Download
2
Embed Size (px)
Citation preview
Lecture Notes on
C��Algebras and Quantum Mechanics
Draft� � April ����
N�P� Landsman
Korteweg�de Vries Institute for Mathematics� University of Amsterdam�Plantage Muidergracht ���
���� TV AMSTERDAM� THE NETHERLANDS
email� nplwinsuvanlhomepage� http���turingwinsuvanl��npl�
telephone� ������� ���o�ce� Euclides ���a
�
CONTENTS �
Contents
� Historical notes ��� Origins in functional analysis and quantum mechanics ��� Rings of operators �von Neumann algebras� ��� Reduction of unitary group representations �� The classi�cation of factors ��� C��algebras ��
� Elementary theory of C��algebras ���� Basic de�nitions ���� Banach algebra basics ���� Commutative Banach algebras ���� Commutative C��algebras ���� Spectrum and functional calculus ��� Positivity in C��algebras ���� Ideals in C��algebras ���� States ���� Representations and the GNS�construction ����� The Gel�fand�Neumark theorem ����� Complete positivity ����� Pure states and irreducible representations ����� The C��algebra of compact operators ����� The double commutant theorem ��
� Hilbert C��modules and induced representations ���� Vector bundles � �� Hilbert C��modules ���� The C��algebra of a Hilbert C��module ��� Morita equivalence ��� Rie�el induction �� The imprimitivity theorem ��� Group C��algebras ��� C��dynamical systems and crossed products ���� Transformation group C��algebras ����� The abstract transitive imprimitivity theorem ����� Induced group representations ����� Mackey�s transitive imprimitivity theorem ��
� Applications to quantum mechanics ���� The mathematical structure of classical and quantum mechanics ���� Quantization ���� Stinespring�s theorem and coherent states ���� Covariant localization in con�guration space � �� Covariant quantization on phase space ��
Literature ��
� CONTENTS
�
� Historical notes
��� Origins in functional analysis and quantum mechanics
The emergence of the theory of operator algebras may be traced back to �at least� three develop�ments
� The work of Hilbert and his pupils in G�ottingen on integral equations� spectral theory� andin�nite�dimensional quadratic forms ��������
� The discovery of quantum mechanics by Heisenberg ������ in G�ottingen and �independently�by Schr�odinger in Z�urich ���� ��
� The arrival of John von Neumann in G�ottingen ���� � to become Hilbert�s assistant
Hilbert�s memoirs on integral equations appeared between ���� and ��� In ���� his student ESchmidt de�ned the space �� in the modern sense F Riesz studied the space of all continuous linearmaps on �� ������� and various examples of L��spaces emerged around the same time However�the abstract concept of a Hilbert space was still missing
Heisenberg discovered a form of quantum mechanics� which at the time was called �matrixmechanics� Schr�odinger was led to a di�erent formulation of the theory� which he called �wavemechanics� The relationship and possible equivalence between these alternative formulations ofquantum mechanics� which at �rst sight looked completely di�erent� was much discussed at thetime It was clear from either approach that the body of work mentioned in the previous paragraphwas relevant to quantum mechanics
Heisenberg�s paper initiating matrix mechanics was followed by the �Dreim�annerarbeit� of Born�Heisenberg� and Jordan ���� �� all three were in G�ottingen at that time Born was one of thefew physicists of his time to be familiar with the concept of a matrix� in previous research hehad even used in�nite matrices �Heisenberg�s fundamental equations could only be satis�ed byin�nite�dimensional matrices� Born turned to his former teacher Hilbert for mathematical adviceHilbert had been interested in the mathematical structure of physical theories for a long time�his Sixth Problem ������ called for the mathematical axiomatization of physics Aided by hisassistants Nordheim and von Neumann� Hilbert thus ran a seminar on the mathematical structureof quantum mechanics� and the three wrote a joint paper on the subject �now obsolete�
It was von Neumann alone who� at the age of ��� saw his way through all structures andmathematical di�culties In a series of papers written between ���������� culminating in his bookMathematische Grundlagen der Quantenmechanik ������� he formulated the abstract concept of aHilbert space� developed the spectral theory of bounded as well as unbounded normal operatorson a Hilbert space� and proved the mathematical equivalence between matrix mechanics and wavemechanics Initiating and largely completing the theory of self�adjoint operators on a Hilbertspace� and introducing notions such as density matrices and quantum entropy� this book remainsthe de�nitive account of the mathematical structure of elementary quantum mechanics �vonNeumann�s book was preceded by Dirac�s The Principles of Quantum Mechanics ������� whichcontains a heuristic and mathematically unsatisfactory account of quantum mechanics in terms oflinear spaces and operators�
��� Rings of operators �von Neumann algebras�
In one of his papers on Hilbert space theory ������� von Neumann de�nes a ring of operatorsM �nowadays called a von Neumann algebra� as a ��subalgebra of the algebra B�H� of allbounded operators on a Hilbert space H �ie� a subalgebra which is closed under the involutionA� A�� that is closed �ie� sequentially complete� in the weak operator topology The latter maybe de�ned by its notion of convergence� a sequence fAng of bounded operators weakly convergesto A when ��� An�� � ��� A�� for all � � H This type of convergence is partly motivated byquantum mechanics� in which ��� A�� is the expectation value of the observable A in the state ��provided that A is self�adjoint and � has unit norm
� HISTORICAL NOTES
For example� B�H� is itself a von Neumann algebra �Since the weak topology is weaker thanthe uniform �or norm� topology on B�H�� a von Neumann algebra is automatically norm�closedas well� so that� in terminology to be introduced later on� a von Neumann algebra becomes a C��algebra when one changes the topology from the weak to the uniform one However� the naturaltopology on a von Neumann algebra is neither the weak nor the uniform one�
In the same paper� von Neumann proves what is still the basic theorem of the subject� a ��subalgebra M of B�H�� containing the unit operator I� is weakly closed i� M�� � M Here thecommutant M� of a collection M of bounded operators consists of all bounded operators whichcommute with all elements of M� and the bicommutant M�� is simply �M��� This theorem isremarkable� in relating a topological condition to an algebraic one� one is reminded of the muchsimpler fact that a linear subspace K of H is closed i� K��� where K� is the orthogonal complementof K in H
Von Neumann�s motivation in studying rings of operators was plurifold His primary motivationprobably came from quantum mechanics� unlike many physicists then and even now� he knewthat all Hilbert spaces of a given dimension are isomorphic� so that one cannot characterize aphysical system by saying that �its Hilbert space of �pure� states is L��R� �� Instead� von Neumannhoped to characterize quantum�mechanical systems by algebraic conditions on the observablesThis programme has� to some extent been realized in algebraic quantum �eld theory �Haag andfollowers�
Among von Neumann�s interest in quantum mechanics was the notion of entropy� he wishedto de�ne states of minimal information When H � C n for n � �� such a state is given by thedensity matrix � � I�n� but for in�nite�dimensional Hilbert spaces this state may no longer bede�ned Density matrices may be regarded as states on the von Neumann algebraB�H� �in thesense of positive linear functionals which map I to �� As we shall see� there are von Neumannalgebras on in�nite�dimensional Hilbert spaces which do admit states of minimal information thatgeneralize I�n� viz the factors of type II� �see below�
Furthermore� von Neumann hoped that the divergences in quantum �eld theory might be re�moved by considering algebras of observables di�erent from B�H� This hope has not materialized�although in algebraic quantum �eld theory the basic algebras of local observables are� indeed� notof the form B�H�� but are all isomorphic to the unique hyper�nite factor of type III� �see below�
Motivation from a di�erent direction came from the structure theory of algebras In the presentcontext� a theorem of Wedderburn says that a von Neumann algebra on a �nite�dimensional Hilbertspace is �isomorphic to� a direct sum of matrix algebras Von Neumann wondered if this� or asimilar result in which direct sums are replaced by direct integrals �see below�� still holds whenthe dimension of H is in�nite �As we shall see� it does not�
Finally� von Neumann�s motivation came from group representations Von Neumann�s bicom�mutant theorem implies a useful alternative characterization of von Neumann algebras� from nowon we add to the de�nition of a von Neumann algebra the condition that M contains I
The commutant of a group U of unitary operators on a Hilbert space is a von Neumann algebra�and� conversely� every von Neumann algebra arises in this way In one direction� one trivially veri�esthat the commutant of any set of bounded operators is weakly closed� whereas the commutant ofa set of bounded operators which is closed under the involution is a ��algebra In the oppositedirection� given M� one takes U to be the set of all unitaries in M�
This alternative characterization indicates why von Neumann algebras are important in physics�the set of bounded operators on H which are invariant under a given group representation U�G�on H is automatically a von Neumann algebra �Note that a given group U of unitaries on H maybe regarded as a representation U of U itself� where U is the identity map�
��� Reduction of unitary group representations
The �possible� reduction of U�G� is determined by the von Neumann algebras U�G��� and U�G��For example� U is irreducible i� U�G�� � C I �Schur�s lemma� The representation U is calledprimary when U�G��� has a trivial center� that is� when U�G��� � U�G�� � C I When G iscompact� so that U is discretely reducible� this implies that U is a multiple of a �xed irreducible
��� Reduction of unitary group representations �
representation U� on a Hilbert space H� � so that H � H� �K� and U � U� � IKWhen G is not compact� but still assumed to be locally compact� unitary representations may
be reducible without containing any irreducible subrepresentation This occurs already in thesimplest possible cases� such as the regular representation of G � R on H � L��R�� that is� oneputs U�x���y� � ��y � x� The irreducible would�be subspaces of H would be spanned by thevectors �p�y� �� exp�ipy�� but these functions do not lie in L��R� The solution to this problemwas given by von Neumann in a paper published in ����� but written in the thirties �the ideas init must have guided von Neumann from at least ��� on�
Instead of decomposing H as a direct sum� one should decompose it as a direct integral�To do so� one needs to assume that H is separable� This means that �rstly one has a measurespace ��� �� and a family of Hilbert spaces fH�g��� A section of this family is a function� � � � fH�g��� for which ���� � H� To de�ne the direct integral of the H� with respect tothe measure �� one needs a sequence of sections f�ng satisfying the two conditions that �rstly thefunction � � ��n�����m����� be measurable for all n�m� and secondly that for each �xed � the�n span H� There then exists a unique maximal linear subspace �� of the space � of all sectionswhich contains all �n� and for which all sections �� �������� are measurable
For ��� � �� it then makes sense to de�ne
����� ��
Z�
d���� ������������ �
The direct integral Z �
�
d����H�
is then by de�nition the subset of �� of functions � for which ����� � � When � is discrete�the direct integral reduces to a direct sum
An operator A on this direct integral Hilbert space is said to be diagonal when
A���� � A�����
for some �suitably measurable� family of operators A� on H� We then write
A �
Z �
�
d����A��
Thus a unitary group representation U�G� on H is diagonal when
U�x����� � U��x���
for all x � G� in which case we� of course� write
U �
Z �
�
d����U��
Reducing a given representation U on some Hilbert space then amounts to �nding a unitary mapV between H and some direct integral Hilbert space� such that each H� carries a representationU�� and V U�x�V � is diagonal in the above sense� with A� � U��x� When H is separable� one mayalways reduce a unitary representation in such a way that the U� occurring in the decompositionare primary� and this central decomposition of U is essentially unique
To completely reduce U � one needs the U� to be irreducible� so that � is the space �G of allequivalence classes of irreducible unitary representations of G Complete reduction therefore callsfor a further direct integral decomposition of primary representations� this will be discussed below
For example� one may take � � R with Lebesgue measure �� and take the sequence f�ng toconsist of a single strictly positive measurable function This leads to the direct integral decom�position
L��R� �
Z �
R
dpHp�
� � HISTORICAL NOTES
in which each Hp is C To reduce the regular representation of R on L��R�� one simply performsa Fourier transform V � L��R� � L��R�� ie�
V ��p� �
ZR
dy e�ipy��y��
This leads to V U�x�V ���p� � exp�ipx���p�� so that U has been diagonalized� the U��x� above arenow the one�dimensional operators Up�x� � exp�ipx� on Hp � C We have therefore completelyreduced U
As far as the reduction of unitary representations is concerned� there exist two radically di�erentclasses of locally compact groups �the class of all locally compact groups includes� for example�all �nite�dimensional Lie groups and all discrete groups� A primary representation is said to beof type I when it may be decomposed as the direct sum of irreducible subrepresentations� thesesubrepresentations are necessarily equivalent A locally compact group is said to be type I ortame when every primary representation is a multiple of a �xed irreducible representation� inother words� a group is type I when all its primary representations are of type I If not� the groupis called non�type I or wild An example of a wild group� well known to von Neumann� is the freegroup on two generators Another example� discovered at a later stage� is the group of matrices ofthe form �� eit � z
� ei�t w� � �
�A �
where � is an irrational real number� t � R� and z� w � C When G is wild� curious phenomena may occur By de�nition� a wild group has primary unitary
representations which contain no irreducible subrepresentations More bizarrely� representationsof the latter type may be decomposed in two alternative ways
U �
Z �
�G
d����U� �
Z �
�G
d����U� �
where the measures �� and �� are disjoint �that is� supported by disjoint subsets of �G�A reducible primary representation U may always be decomposed as U � Uh � Uh In case
that U is not equivalent to Uh� and U is not of type I� it is said to be a representation of type IIWhen U is neither of type I nor of type II� it is of type III In that case U is equivalent to Uh�indeed� all �proper� subrepresentations of a primary type III representation are equivalent
��� The classi�cation of factors
Between ��� and ���� von Neumann wrote � lengthy� di�cult� and profound papers �� of whichwere in collaboration with Murray� in which the study of his �rings of operators� was initiated�According to IE Segal� these papers form �perhaps the most original major work in mathematicsin this century��
The analysis of Murray and von Neumann is based on the study of the projections in a vonNeumann algebra M �a projection is an operator p for which p� � p� � p�� indeed� M is generatedby its projections They noticed that one may de�ne an equivalence relation on the set of allprojections in M� in which p q i� there exists a partial isometry V in M such that V �V � pand V V � � q When M B�H�� the operator V is unitary from pH to qH� and annihilates pH�Hence when M � B�H� one has p q i� pH and qH have the same dimension� for in that caseone may take any V with the above properties
An equivalent characterization of arises when we write M � U�G�� for some unitary represen�tation U of a group G �as we have seen� this always applies�� then p q i� the subrepresentationspU and qU �on pH and qH� respectively�� are unitarily equivalent
Moreover� Murray and von Neumann de�ne a partial orderering on the collection of all pro�jections in M by declaring that p � q when pq � p� that is� when pH qH This induces apartial orderering on the set of equivalence classes of projections by putting �p � �q when the
��� The classi�cation of factors �
equivalence classes �p and �q contain representatives !p and !q such that !p � !q For M � B�H�this actually de�nes a total ordering on the equivalence classes� in which �p � �q when pH hasthe same dimension as qH� as we just saw� this is independent of the choice of p � �p and q � �q
More generally� Murray and von Neumann showed that the set of equivalence classes of projec�tions in M is totally ordered by � whenever M is a factor A von Neumann algebra M is a factorwhen M �M� � C I� when M � U�G�� this means that M is a factor i� the representation U isprimary The study of von Neumann algebras acting on separable Hilbert spaces H reduces to thestudy of factors� for von Neumann proved that every von Neumann algebra M B�H� may beuniquely decomposed� as in
H �
Z �
�
d����H��
M �
Z �
�
d����M��
where �almost� each M� is a factor For M � U�G�� the decomposition of H amounts to thecentral decomposition of U�G�
As we have seen� for the factor M � B�H� the dimension d of a projection is a completeinvariant� distinguishing the equivalence classes �p The dimension is a function from the set ofall projections in B�H� to R� ��� satisfying
� d�p� � when p � �� and d��� � ��
� d�p� � d�q� i� �p �q �
� d�p " q� � d�p� " d�q� when pq � � �ie� when pH and qH are orthogonal�
� d�p� �� i� p is �nite
Here a projection in B�H� is called �nite when pH is �nite�dimensional Murray and von Neumannnow proved that on any factor M �acting on a separable Hilbert space� there exists a function dfrom the set of all projections in M to R� � �� satisfying the above properties Moreover� dis unique up to �nite rescaling For this to be the possible� Murray and von Neumann de�ne aprojection to be nite when it is not equivalent to any of its �proper� sub�projections� an inniteprojection is then a projection which has proper sub�projections to which it is equivalent ForM � B�H� this generalized notion of �niteness coincides with the usual one� but in other factorsall projections may be in�nite in the usual sense� yet some are �nite in the sense of Murray andvon Neumann One may say that� in order to distinguish in�nite�dimensional but inequivalentprojections� the dimension function d is a �renormalized� version of the usual one
A �rst classi�cation of factors �on a separable Hilbert space� is now performed by consideringthe possible �niteness of its projections and the range of d A projection p is called minimal oratomic when there exists no q � p �ie� q � p and q � p� One then has the following possibilitiesfor a factor M
� type In� where n � �� M has minimal projections� all projections are �nite� and d takesthe values f�� �� � � � � ng A factor of type In is isomorphic to the algebra of n� n matrices
� type I�� M has minimal projections� and d takes the values f�� �� � � � ��g Such a factor isisomorphic to B�H� for separable in�nite�dimensional H
� type II�� M has no minimal projections� all projections are in�nite�dimensional in the usualsense� and I is �nite Normalizing d such that d�I� � �� the range of d is the interval ��� �
� type II�� M has no minimal projections� all nonzero projections are in�nite�dimensionalin the usual sense� but M has �nite�dimensional projections in the sense of Murray and vonNeumann� and I is in�nite The range of d is ����
�� � HISTORICAL NOTES
� type III� M has no minimal projections� all nonzero projections are in�nite�dimensionaland equivalent in the usual sense as well as in the sense of Murray and von Neumann� and dassumes the values f���g
With M � U�G��� where� as we have seen� the representation U is primary i� M is a factor� Uis of a given type i� M is of the same type
One sometimes says that a factor is nite when I is �nite �so that d�I� � ��� hence type Inand type II� factors are �nite Factors of type I� and II� are then called seminite� and typeIII factors are purely innite
It is hard to construct an example of a II� factor� and even harder to write down a type IIIfactor Murray and von Neumann managed to do the former� and von Neumann did the latterby himself� but only � years after he and Murray had recognized that the existence of type IIIfactors was a logical possibility However� they were unable to provide a further classi�cation ofall factors� and they admitted having no tools to study type III factors
Von Neumann was fascinated by II� factors In view of the range of d� he believed thesede�ned some form of continuous geometry Moreover� the existence of a II� factor solved one ofthe problems that worried him in quantum mechanics For he showed that on a II� factor Mthe dimension function d� de�ned on the projections in M� may be extended to a positive linearfunctional tr on M� with the property that tr�UAU�� � tr�A� for all A �M and all unitaries U inM This �trace� satis�es tr�I� � d�I� � �� and gave von Neumann the state of minimal informationhe had sought Partly for this reason he believed that physics should be described by II� factors
At the time not many people were familiar with the di�cult papers of Murray and von Neu�mann� and until the sixties only a handful of mathematicians worked on operator algebras �eg�Segal� Kaplansky� Kadison� Dixmier� Sakai� and others� The precise connection between von Neu�mann algebras and the decomposition of unitary group representations envisaged by von Neumannwas worked out by Mackey� Mautner� Godement� and Adel�son�Vel�skii
In the sixties� a group of physicists� led by Haag� realized that operator algebras could be a usefultool in quantum �eld theory and in the quantum statistical mechanics of in�nite systems This hasled to an extremely fruitful intercation between physics and mathematics� which has helped bothsubjects In particular� in ���� Haag observed a formal similarity between the collection of allvon Neumann algebras on a Hilbert space and the set of all causally closed subsets of Minkowksispace�time Here a region O in space�time is said to be causally closed when O�� � O� whereO� consists of all points that are spacelike separated from O The operation O � O� on causallyclosed regions in space�time is somewhat analogous to the operation M � M� on von Neumannalgebras Thus Haag proposed that a quantum �eld theory should be de�ned by a net of localobservables� this is a map O � M�O� from the set of all causally closed regions in space�timeto the set of all von Neumann algebras on some Hilbert space� such that M�O�� M�O�� whenO� O�� and M�O�� � M�O��
This idea initiated algebraic quantum eld theory� a subject that really got o� the groundwith papers by Haag�s pupil Araki in �� � and by Haag and Kastler in �� � From then tillthe present day� algebraic quantum �eld theory has attracted a small but dedicated group ofmathematical physicists One of the result has been that in realistic quantum �eld theories thelocal algebras M�O� must all be isomorphic to the unique hyper�nite factor of type III� discussedbelow �Hence von Neumann�s belief that physics should use II� factors has not been vindicated�
A few years later ��� ��� an extraordinary coincidence took place� which was to play an es�sential role in the classi�cation of factors of type III On the mathematics side� Tomita developeda technique in the study of von Neumann algebras� which nowadays is called modular theoryor Tomita�Takesaki theory �apart from clarifying Tomita�s work� Takesaki made essential con�tributions to this theory� Among other things� this theory leads to a natural time�evolution oncertain factors On the physics side� Haag� Hugenholtz� and Winnink characterized states of ther�mal equilibrium of in�nite quantum systems by an algebraic condition that had previously beenintroduced in a heuristic setting by Kubo� Martin� and Schwinger� and is therefore called the KMScondition This condition leads to type III factors equipped with a time�evolution which coincidedwith the one of the Tomita�Takesaki theory
In the hands of Connes� the Tomita�Takesaki theory and the examples of type III factors
��� C��algebras ��
provided by physicists �Araki� Woods� Powers� and others� eventually led to the classi�cation of allhypernite factors of type II and III �the complete classi�cation of all factors of type I is alreadygiven by the list presented earlier� These are factors containing a sequence of �nite�dimensionalsubalgebras M� � M� � � � � M� such that M is the weak closure of �nMn �Experience showsthat all factors playing a role in physics are hyper�nite� and many natural examples of factorsconstructed by purely mathematical techniques are hyper�nite as well� The work of Connes� forwhich he was awarded the Fields Medal in ����� and others� led to the following classi�cation ofhyper�nite factors of type II and III �up to isomorphism��
� There is a unique hyper�nite factor of type II� �In physics this factor occurs when oneconsiders KMS�states at in�nite temperature�
� There is a unique hyper�nite factor of type II�� namely the tensor product of the hyper�niteII��factor with B�K�� for an in�nite�dimensional separable Hilbert space K
� There is a family of type III factors� labeled by � � ��� � For � � � the factor of type III�is unique There is a family of type III� factors� which in turn is has been classi�ed in termsof concepts from ergodic theory
As we have mentioned already� the unique hyper�nite III� factor plays a central role in algebraicquantum �eld theory The unique hyper�nite II� factor was crucial in a spectacular development�in which the theory of inclusions of II� factors was related to knot theory� and even led to a newknot invariant In ���� Jones was awarded a Fields medal for this work� the second one to be givento the once obscure �eld of operator algebras
��� C�algebras
In the midst of the Murray�von Neumann series of papers� Gel�fand initiated a separate develop�ment� combining operator algebras with the theory of Banach spaces In ���� he de�ned the con�cept of a Banach algebra� in which multiplication is �separately� continuous in the norm�topologyHe proceeded to de�ne an intrinsic spectral theory� and proved most basic results in the theory ofcommutative Banach algebras
In ���� Gel�fand and Neumark de�ned what is now called a C��algebra �some of their axiomswere later shown to be super#uous�� and proved the basic theorem that each C��algebra is isomor�phic to the norm�closed ��algebra of operators on a Hilbert space Their paper also contained therudiments of what is now called the GNS construction� connecting states to representations Inits present form� this construction is due to Segal ������� a great admirer of von Neumann� whogeneralized von Neumann�s idea of a state as a positive normalized linear functional from B�H� toarbitrary C��algebras Moreover� Segal returned to von Neumann�s motivation of relating operatoralgebras to quantum mechanics
As with von Neumann algebras� the sixties brought a fruitful interaction between C��algebrasand quantum physics Moreover� the theory of C��algebras turned out to be interesting both forintrinsic reasons �structure and representation theory of C��algebras�� as well as because of itsconnections with a number of other �elds of mathematics Here the strategy is to take a givenmathematical structure� try and �nd a C��algebra which encodes this structure in some way� andthen obtain information about the structure through proving theorems about the C��algebra ofthe structure
The �rst instance where this led to a deep result which has not been proved in any otherway is the theorem of Gel�fand and Raikov ������� stating that the unitary representations of alocally compact group separate the points of the group �that is� for each pair x � y there exists aunitary representation U for which U�x� � U�y� This was proved by constructing a C��algebraC��G� of the group G� showing that representations of C��G� bijectively correspond to unitaryrepresentations of G� and �nally showing that the states of an arbitrary C��algebra A separate theelements of A
Other examples of mathematical structures that may be analyzed through an appropriate C��algebra are group actions� groupoids� foliations� and complex domains The same idea lies at the
�� � ELEMENTARY THEORY OF C��ALGEBRAS
basis of non�commutative geometry and non�commutative topology Here the startingpoint is another theorem of Gel�fand� stating that any commutative C��algebra �with unit� isisomorphic to C�X�� where X is a compact Hausdor� space The strategy is now that the basic toolsin the topology of X � and� when appropriate� in its di�erential geometry� should be translated intotools pertinent to the C��algebra C�X�� and that subsequently these tools should be generalizedto non�commutative C��algebras
This strategy has been successful in K�theory� whose non�commutative version is even simplerthan its usual incarnation� and in �de Rham� cohomology theory� whose non�commutative versionis called cyclic cohomology Finally� homology� cohomology� K�theory� and index theory havenbeen uni�ed and made non�commutative in the KK�theory of Kasparov The basic tool in KK�theory is the concept of a Hilbert C��module� which we will study in detail in these lectures
� Elementary theory of C��algebras
��� Basic de�nitions
All vector spaces will be de�ned over C � and all functions will be C �valued� unless we explicitlystate otherwise The abbreviation �i�� means �if and only if�� which is the same as the symbol �An equation of the type a �� b means that a is by de�nition equal to b
Denition ��� A norm on a vector space V is a map k k � V � R such that
�� k v k� � for all v � V��� k v k� � i� v � ��
�� k �v k� j�j k v k for all � � C and v � V��� k v " w k�k v k " k w k triangle inequality�
A norm on V de�nes a metric d on V by d�v� w� ��k v � w k� A vector space with a norm whichis complete in the associated metric in the sense that every Cauchy sequence converges is calleda Banach space� We will denote a generic Banach space by the symbol B�
The two main examples of Banach spaces we will encounter are Hilbert spaces and certaincollections of operators on Hilbert spaces
Denition ��� A pre�inner product on a vector space V is a map � � � � V �V � C such that
�� ���v� " ��v�� ��w� " ��w�� � �����v�� w�� " �����v�� w�� " �����v�� w�� " �����v�� w�� forall ��� ��� ��� �� � C and v�� v�� w�� w� � V�
�� �v� v� � � for all v � V�An equivalent set of conditions is
�� �v� w� � �w� v� for all v� w � V��� �v� ��w� " ��w�� � ���v� w�� " ���v� w�� for all ��� �� � C and v� w�� w� � V��� �v� v� � � for all v � V�
A pre�inner product for which �v� v� � � i� v � � is called an inner product�
The equivalence between the two de�nitions of a pre�inner product is elementary� in fact� toderive the �rst axiom of the second characterization from the �rst set of conditions� it is enoughto assume that �v� v� � R for all v �use this reality with v � v " iw� Either way� one derives theCauchy�Schwarz inequality
j�v� w�j� � �v� v��w�w�� ����
��� Basic de�nitions ��
for all v� w � V Note that this inequality is valid even when � � � is not an inner product� butmerely a pre�inner product
It follows from these properties that an inner product on V de�nes a norm on V by k v k��p�v� v�� the triangle inequality is automatic
Denition ��� A Hilbert space is a vector space with inner product which is complete in theassociated norm� We will usually denote Hilbert spaces by the symbol H�
A Hilbert space is completely characterized by its dimension �ie� by the cardinality of anarbitrary orthogonal basis� To obtain an interesting theory� one therefore studies operators on aHilbert space� rather than the Hilbert space itself To obtain a satisfactory mathematical theory� itis wise to restrict oneself to bounded operators We recall this concept in the more general contextof arbitrary Banach spaces
Denition ��� A bounded operator on a Banach space B is a linear map A � B � B forwhich
k A k�� sup fk Av k j v � B� k v k� �g ��� ����
The number k A k is the operator norm� or simply the norm� of A This terminology isjusti�ed� as it follows almost immediately from its de�nition �and from the properties of the thenorm on B� that the operator norm is indeed a norm
�It is easily shown that a linear map on a Banach space is continuous i� it is a bounded operator�but we will never use this result Indeed� in arguments involving continuous operators on a Banachspace one almost always uses boundedness rather than continuity�
When B is a Hilbert space H the expression ���� becomes
k A k�� sup f�A�� A���
� j� � H� ����� � �g� ����
When A is bounded� it follows that
k Av k�k A k k v k ����
for all v � B Conversely� when for A � � there is a C � such that k Av k� C k v k for all v�then A is bounded� with operator norm k A k equal to the smallest possible C for which the aboveinequality holds
Proposition ��� The space B�B� of all bounded operators on a Banach space B is itself aBanach space in the operator norm�
In view of the comments following ����� it only remains to be shown that B�B� is complete inthe operator norm Let fAng be a Cauchy sequence in B�B� In other words� for any � � thereis a natural number N��� such that k An � Am k� � when n�m N��� For arbitrary v � B� thesequence fAnvg is a Cauchy sequence in B� because
k Anv �Amv k�k An �Am k k v k� � k v k ����
for n�m N��� Since B is complete by assumption� the sequence fAnvg converges to some w � BNow de�ne a map A on B by Av �� w � limnAnv This map is obviously linear Taking n � �in ����� we obtain
k Av �Amv k� � k v k �� �
for all m N��� and all v � B It now follows from ���� that A � Am is bounded SinceA � �A�Am� "Am� and B�B� is a linear space� we infer that A is bounded Moreover� �� � and���� imply that k A � Am k� � for all m N���� so that fAng converges to A Since we havejust seen that A � B�B�� this proves that B�B� is complete �
We de�ne a functional on a Banach space B as a linear map � � B � C which is continuousin that ��v�j � C k v k for some C� and all v � B The smallest such C is the norm
k � k�� sup fj��v�j� v � B� k v k� �g� ����
�� � ELEMENTARY THEORY OF C��ALGEBRAS
The dual B� of B is the space of all functionals on B Similarly to the proof of ���� one shows thatB� is a Banach space For later use� we quote� without proof� the fundamental Hahn�Banachtheorem
Theorem ��� For a functional �� on a linear subspace B� of a Banach space B there exists afunctional � on B such that � � �� on B� and k � k�k �� k� In other words each functional de�nedon a linear subspace of B has an extension to B with the same norm�
Corollary ��� When ��v� � � for all � � B� then v � ��
For v � � we may de�ne a functional �� on C v by ����v� � �� and extend it to a functional �on B with norm � �
Recall that an algebra is a vector space with an associative bilinear operation ��multiplication��� � A� A � A� we usually write AB for A � B It is clear that B�B� is an algebra under operatormultiplication Moreover� using ���� twice� for each v � B one has
k ABv k�k A k k Bv k�k A k k B k k v k �Hence from ���� we obtain k AB k�k A k k B kDenition ��� A Banach algebra is a Banach space A which is at the same time an algebra in which for all A�B � A one has
k AB k�k A k k B k � ����
It follows that multiplication in a Banach algebra is separately continuous in each variableAs we have just seen� for any Banach space B the space B�B� of all bounded operators on B is
a Banach algebra In what follows� we will restrict ourselves to the case that B is a Hilbert spaceH� this leads to the Banach algebra B�H� This algebra has additional structure
Denition ��� An involution on an algebra A is a real�linear map A � A� such that for allA�B � A and � � C one has
A�� � A� ����
�AB�� � B�A�� �����
��A�� � �A�� �����
A ��algebra is an algebra with an involution�
The operator adjoint A� A� on a Hilbert space� de�ned by the property ��� A��� �� �A�����de�nes an involution on B�H� Hence B�H� is a ��algebra As in this case� an element A of a C��algebraA is called self�adjoint when A� � A� we sometimes denote the collection of all self�adjointelements by
AR �� fA � AjA� � Ag� �����
Since one may write
A � A� " iA�� ��A " A�
�" i
A�A�
�i� �����
every element of A is a linear combination of two self�adjoint elementsTo see how the norm in B�H� is related to the involution� we pick � � H� and use the Cauchy�
Schwarz inequality and ���� to estimate
k A� k�� �A�� A�� � ��� A�A�� �k � k k A�A� k�k A�A k k � k� �Using ���� and ����� we infer that
k A k��k A�A k�k A� k k A k � �����
This leads to k A k�k A� k Replacing A by A� and using ���� yields k A� k�k A k� so thatk A� k�k A k Substituting this in ������ we derive the crucial property k A�A k�k A k�
This motivates the following de�nition
��� Banach algebra basics ��
Denition ���� A C��algebra is a complex Banach space A which is at the same time a ��algebra such that for all A�B � A one has
k AB k � k A k k B k� �����
k A�A k � k A k� � ��� �
In other words a C��algebra is a Banach ��algebra in which ���� holds�
Here a Banach ��algebra is� of course� a Banach algebra with involution Combining ��� � and������ one derives k A k�k A� k� as in the preceding paragraph� we infer that for all elements Aof a C��algebra one has the equality
k A� k�k A k � �����
The same argument proves the following
Lemma ���� A Banach ��algebra in which k A k��k A�A k is a C��algebra�
We have just shown that B�H� is a C��algebra Moreover� each �operator� norm�closed ��algebra in B�H� is a C��algebra by the same argument A much deeper result� which we willformulate precisely and prove in due course� states the converse of this� each C��algebra is isomor�phic to a norm�closed ��algebra in B�H�� for some Hilbert space H Hence the axioms in ����characterize norm�closed ��algebras on Hilbert spaces� although the axioms make no reference toHilbert spaces at all
For later use we state some self�evident de�nitions
Denition ���� Amorphism between C��algebras A�B is a complex� linear map � � A� B
such that
��AB� � ��A���B�� �����
��A�� � ��A�� �����
for all A�B � A� An isomorphism is a bijective morphism� Two C��algebras are isomorphicwhen there exists an isomorphism between them�
One immediately checks that the inverse of a bijective morphism is a morphism It is remark�able� however� that an injective morphism �and hence an isomorphism� between C��algebras isautomatically isometric For this reason the condition that an isomorphism be isometric is notincluded in the de�nition
��� Banach algebra basics
The material in this section is not included for its own interest� but because of its role in thetheory of C��algebras Even in that special context� it is enlightening to see concepts such as thespectrum in their general and appropriate setting
Recall De�nition ��� A unit in a Banach algebra A is an element I satisfying IA � AI� Afor all A � A� and
k Ik� �� �����
A Banach algebra with unit is called unital We often write z for zI� where z � C Note that ina C��algebra the property IA � AI � A already implies� ������ take A � I�� so that I�I � I��taking the adjoint� this implies I� � I� so that ����� follows from ��� �
When a Banach algebra A does not contain a unit� we can always add one� as follows Formthe vector space
AI �� A� C � �����
and make this into an algebra by means of
�A " �I��B" �I� �� AB " �B " �A " ��I� �����
� � ELEMENTARY THEORY OF C��ALGEBRAS
where we have written A" �I for �A� ��� etc In other words� the number � in C is identi�ed withI Furthermore� de�ne a norm on AI by
k A " �Ik��k A k "j�j� �����
In particular� k Ik� � Using ����� in A� as well as ����� one sees from ����� and ����� that
k �A " �I��B" �I� k�k A k k B k "j�j k B k "j�j k A k "j�j j�j �k A " �Ik k B " �Ik�
so that AI is a Banach algebra with unit Since by ����� the norm of A � A in A coincides withthe norm of A " �I in AI� we have shown the following
Proposition ��� For every Banach algebra without unit there exists a unital Banach algebra AIand an isometric hence injective morphism A� AI such that AI�A � C �
As we shall see at the end of section ��� the unitization AI with the given properties is notunique
Denition ��� Let A be a unital Banach algebra� The resolvent ��A� of A � A is the set ofall z � C for which A� zI has a two�sided inverse in A�
The spectrum �A� of A � A is the complement of ��A� in C � in other words �A� is the setof all z � C for which A� zI has no two�sided inverse in A�
When A has no unit the resolvent and the spectrum are de�ned through the embedding of A inAI� A� C �
When A is the algebra of n � n matrices� the spectrum of A is just the set of eigenvaluesFor A � B�H�� De�nition ��� reproduces the usual notion of the spectrum of an operator on aHilbert space
When A has no unit� the spectrum �A� of A � A always contains zero� since it follows from����� that A never has an inverse in AI
Theorem ��� The spectrum �A� of any element A of a Banach algebra is
�� contained in the set fz � C j jzj � k A kg��� compact�
�� not empty�
The proof uses two lemmas We assume that A is unital
Lemma ��� When k A k� � the sumPn
k��Ak converges to �I�A����
Hence �A� zI��� always exists when jzj k A k�We �rst show that the sum is a Cauchy sequence Indeed� for n m one has
knX
k��
Ak �mXk��
Ak k�knX
k�m��
Ak k�nX
k�m��
k Ak k�nX
k�m��
k A kk �
For n�m�� this goes to � by the theory of the geometric series Since A is complete� the Cauchysequence
Pnk�� A
k converges for n�� Now compute
nXk��
Ak�I�A� �
nXk��
�Ak �Ak��� � I�An���
Hence
k I�nX
k��
Ak�I�A� k�k An�� k�k A kn���
��� Banach algebra basics ��
which � � for n��� as k A k� � by assumption Thus
limn��
nXk��
Ak�I�A� � I�
By a similar argument�
limn��
�I�A�
nXk��
Ak � I�
so that� by continuity of multiplication in a Banach algebra� one �nally has
limn��
nXk��
Ak � �I�A���� �����
The second claim of the lemma follows because �A � z��� � �z���I� A�z���� which existsbecause k A�z k� � when jzj k A k �
To prove that �A� is compact� it remains to be shown that it is closed
Lemma ��� The setG�A� �� fA � AjA�� existsg �����
of invertible elements in A is open in A�
Given A � G�A�� take a B � A for which k B k� k A�� k�� By ���� this implies
k A��B k�k A�� k k B k� �� ��� �
Hence A"B � A�I"A��B� has an inverse� namely �I"A��B���A��� which exists by ��� � andLemma ��� It follows that all C � A for which k A�C k� � lie in G�A�� for � �k A�� k�� �
To resume the proof of Theorem ���� given A � A we now de�ne a function f � C � A byf�z� �� z�A Since k f�z"���f�z� k� �� we see that f is continuous �take � � � in the de�nitionof continuity� Because G�A� is open in A by Lemma ���� it follows from the topological de�nitionof a continuous function that f���G�A�� is open in A But f���G�A�� is the set of all z � C wherez � A has an inverse� so that f���G�A�� � ��A� This set being open� its complement �A� isclosed
Finally� de�ne g � ��A� � A by g�z� �� �z �A��� For �xed z� � ��A�� choose z � C such thatjz � z�j � k �A � z��
�� k�� From the proof of Lemma ���� with A � A � z� and C � A � z�we see that z � ��A�� as k A� z� � �A� z� k� jz � z�j Moreover� the power series
�
z� �A
nXk��
�z� � z
z� �A
�converges for n�� by Lemma ���� because
k �z� � z��z� �A��� k� jz� � zj k �z� �A��� k� ��
By Lemma ���� the limit n�� of this power series is
�
z� �A
�Xk��
�z� � z
z� �A
��
�
z� � A
����z� � z
z� �A
�����
�
z �A� g�z��
Hence
g�z� �
�Xk��
�z� � z�k�z� �A�k�� �����
�� � ELEMENTARY THEORY OF C��ALGEBRAS
is a norm�convergent power series in z For z � � we write k g�z� k� jzj�� k �I� A�z��� k andobserve that limz�� I�A�z � I� since limz�� k A�z k� � by ���� Hence limz���I�A�z��� �I� and
limz��
k g�z� k� �� �����
Let � � A� be a functional onA� since � is bounded� ����� implies that the function g� � z � ��g�z��is given by a convergent power series� and ����� implies that
limz��
g��z� � �� �����
Now suppose that �A� � �� so that ��A� � C The function g� and hence g�� is then de�ned onC � where it is analytic and vanishes at in�nity In particular� g� is bounded� so that by Liouville�stheorem it must be constant By ����� this constant is zero� so that g � � by Corollary ���This is absurd� so that ��A� � C hence �A� � � �
The fact that the spectrum is never empty leads to the following Gel fand�Mazur theorem�which will be essential in the characterization of commutative C��algebras
Corollary ��� If every element except � of a unital Banach algebra A is invertible then A � Cas Banach algebras�
Since �A� � �� for each A � � there is a zA � C for which A � zAI is not invertible HenceA � zAI � � by assumption� and the map A � zA is the desired algebra isomorphism Sincek A k�k zIk� jzj� this isomorphism is isometric �
De�ne the spectral radius r�A� of A � A by
r�A� �� supfjzj� z � �A�g� �����
From Theorem ���� one immediately infers
r�A� �k A k � �����
Proposition ��� For each A in a unital Banach algebra one has
r�A� � limn��
k An k��n � �����
By Lemma ���� for jzj k A k the function g in the proof of Lemma ��� has the norm�convergent power series expansion
g�z� ��
z
�Xk��
�A
z
�k� �����
On the other hand� we have seen that for any z � ��A� one may �nd a z� � ��A� such that thepower series ����� converges If jzj r�A� then z � ��A�� so ����� converges for jzj r�A� Atthis point the proof relies on the theory of analytic functions with values in a Banach space� whichsays that� accordingly� ����� is norm�convergent for jzj r�A�� uniformly in z Comparing with������ this sharpens what we know from Lemma ��� The same theory says that ����� cannotnorm�converge uniformly in z unless k An k �jzjn � � for large enough n This is true for all z forwhich jzj r�A�� so that
lim supn��
k A k��n� r�A�� �����
To derive a second inequality we use the following polynomial spectral mapping property
Lemma ��� For a polynomial p on C de�ne p� �A�� as fp�z�j z � �A�g� Then
p� �A�� � �p�A��� �����
��� Banach algebra basics ��
To prove this equality� choose z� � � C and compare the factorizations
p�z�� � � c
nYi��
�z � �i�����
p�A�� �I � cnYi��
�A� �i���I�� ��� �
Here the coe�cients c and �i��� are determined by p and � When � � ��p�A�� then p�A�� �I isinvertible� which implies that all A � �i���Imust be invertible Hence � � �p�A�� implies thatat least one of the A � �i���I is not invertible� so that �i��� � �A� for at least one i Hencep��i���� � � � �� ie� � � p� �A�� This proves the inclusion �p�A�� p� �A��
Conversely� when � � p� �A�� then � � p�z� for some z � �A�� so that for some i one musthave �i��� � z for this particular z Hence �i��� � �A�� so that A � �i��� is not invertible�implying that p�A�� �I is not invertible� so that � �p�A�� This shows that p� �A�� �p�A���and ����� follows �
To conclude the proof of Proposition ���� we note that since �A� is closed there is an � � �A�for which j�j � r�A� Since �n � �An� by Lemma ���� one has j�nj � k An k by ����� Hencek An k��n� j�j � r�A� Combining this with ����� yields
lim supn��
k A k��n� r�A� �k An k��n �
Hence the limit must exist� and
limn��
k A k��n � infnk An k��n � r�A�� ������
Denition ��� An ideal in a Banach algebra A is a closed linear subspace I A such thatA � I implies AB � I and BA � I for all B � A�
A left�ideal of A is a closed linear subspace I for which A � I implies BA � I for all B � A�A right�ideal of A is a closed linear subspace I for which A � I implies AB � I for all B � A�A maximal ideal is an ideal I � A for which no ideal !I � A !I � I exists which contains I�
In particular� an ideal is itself a Banach algebra An ideal I that contains an invertible elementA must coincide with A� since A��A � I must lie in I� so that all B � BImust lie in I Thisshows the need for considering Banach algebras with and without unit� it is usually harmless toadd a unit to a Banach algebra A� but a given proper ideal I � A does not contain I� and onecannot add I to I without ruining the property that it is a proper ideal
Proposition ���� If I is an ideal in a Banach algebra A then the quotient A�I is a Banachalgebra in the norm
k ��A� k�� infJ�I
k A " J k �����
and the multiplication��A���B� �� ��AB�� �����
Here � � A� A�I is the canonical projection� If A is unital then A�I is unital with unit ��I��
We omit the standard proof that A�I is a Banach space in the norm ����� As far as theBanach algebra structure is concerned� �rst note that ����� is well de�ned� when J�� J� � I onehas
��A " J����B " J�� � ��AB " AJ� " J�B " J�J�� � ��AB� � ��A���B��
since AJ� " J�B " J�J� � I by de�nition of an ideal� and ��J� � � for all J � I To prove �����observe that� by de�nition of the in�mum� for given A � A� for each � � there exists a J � Isuch that
k ��A� k "� �k A " J k � �����
�� � ELEMENTARY THEORY OF C��ALGEBRAS
For if such a J would not exist� the norm in A�I could not be given by ����� On the other hand�for any J � I it is clear from ����� that
k ��A� k�k ��A " J� k�k A " J k � �����
For A�B � A choose � � and J�� J� � I such that ����� holds for A�B� and estimate
k ��A���B� k � k ��A " J����B " J�� k�k ���A " J���B " J��� k� k �A " J���B " J�� k�k A " J� k k B " J� k� �k ��A� k "���k ��B� k "��� �����
Letting �� � yields k ��A���B� k�k ��A� k k ��B� kWhen A has a unit� it is obvious from ����� that ��I� is a unit in A�I By ����� with A � I
one has k ��I� k�k I k� � On the other hand� from ���� with B � I one derives k ��I� k� �Hence k ��I� k� � �
��� Commutative Banach algebras
We now assume that the Banach algebra A is commutative �that is� AB � BA for all A�B � A�
Denition ��� The structure space $�A� of a commutative Banach algebra A is the set ofall nonzero linear maps � � A� C for which
��AB� � ��A���B� �����
for all A�B � A� We say that such an � is multiplicative�In other words $�A� consists of all nonzero homomorphisms from A to C �
Proposition ��� Let A have a unit I�
�� Each � � $�A� satis�es��I� � �� �����
�� each � � $�A� is continuous with norm
k � k� �� �����
hencej��A�j � k A k ��� �
for all A � A�The �rst claim is obvious� since ��IA� � ��I���A� � ��A�� and there is an A for which
��A� � � because � is not identically zeroFor the second� we know from Lemma ��� that A� z is invertible when jzj k A k� so that
��A � z� � ��A� � z � �� since � is a homomorphism Hence j��A�j � jzj for jzj A k� and��� � follows �
Theorem ��� Let A be a unital commutative Banach algebra� There is a bijective correspon�dence between $�A� and the set of all maximal ideals in A in that the kernel ker��� of each� � $�A� is a maximal ideal I� each maximal ideal is the kernel of some � � $�A� and �� � ��i� I�� � I�� �
The kernel of each � � $�A� is closed� since � is continuous by ���� Furthermore� ker���is an ideal since � satis�es ����� The kernel of every linear map � � V � C on a vector space Vhas codimension one �that is� dim�V� ker���� � ��� so that ker��� is a maximal ideal Again onany vector space� when ker���� � ker���� then �� is a multiple of �� For �i � $�A� this implies�� � �� because of �����
��� Commutative Banach algebras ��
We now show that every maximal ideal I of A is the kernel of some � � $�A� Since I � A�there is a nonzero B � A which is not in I Form
IB �� fBA " J j� A � A� J � Ig�This is clearly a left�ideal� since A is commutative� IB is even an ideal Taking A � � we seeI IB Taking A � I and J � � we see that B � IB � so that IB � I Hence IB � A� as I ismaximal In particular� I� IB � hence I� BA " J for suitable A � A� J � I Apply the canonicalprojection � � A� A�I to this equation� giving
��I� � I� ��BA� � ��B���A��
because of ����� and ��J� � � hence ��A� � ��B��� in A�I Since B was arbitrary �thoughnonzero�� this shows that every nonzero element of A�I is invertible By Corollary �� this yieldsA�I � C � so that there is a homomorphism � � A�I � C Now de�ne a map � � A � C by��A� �� ����A�� This map is clearly linear� since � and � are Also�
��A���B� � ����A������B�� � ����A���B�� � ����AB�� � ��AB��
because of ����� and the fact that � is a homomorphismTherefore� � is multiplicative� it is nonzero because ��B� � �� or because ��I� � � Hence
� � $�A� Finally� I ker��� since I � ker���� but if B �� I we saw that ��B� � �� so thatactually I � ker��� �
By ���� we have $�A� � A� Recall that the weak��topology� also called w��topology�on the dual B� of a Banach space B is de�ned by the convergence �n � � i� �n�v� � ��v� for allv � B The Gel fand topology on $�A� is the relative w��topology
Proposition ��� The structure space $�A� of a unital commutative Banach algebra A is com�pact and Hausdor� in the Gel�fand topology�
The convergence �n � � in the w��topology by de�nition means that �n�A� � ��A� for allA � A When �n � $�A� for all n� one has
j��AB�� ��A���B�j � j��AB�� �n�AB� " �n�A��n�B�� ��A���B�j� j��AB� � �n�AB�j " j�n�A��n�B�� ��A���B�j�
In the second term we write
�n�A��n�B�� ��A���B� � ��n�A�� ��A���n�B� " ��A���n�B�� ��B���
By ��� � and the triangle inequality� the absolute value of the right�hand side is bounded by
k B k j�n�A�� ��A�j" k A k j�n�B�� ��B�j�All in all� when �n � � in the w��topology we obtain j��AB�� ��A���B�j � �� so that the limit� � $�A� Hence $�A� is w��closed
From ����� we have $�A� � A�� �the unit ball in A�� consisting of all functionals with norm� �� By the Banach�Alaoglu theorem� the unit ball in A� is w��compact Being a closed subsetof this unit ball� $�A� is w��compact Since the w��topology is Hausdor� �as is immediate fromits de�nition�� the claim follows �
We embed A in A�� by A� �A� where
�A��� �� ��A�� �����
When � � $�A�� this de�nes �A as a function on $�A� By elementary functional analysis� the w��topology on A� is the weakest topology for which all �A� A � A� are continuous This implies that
�� � ELEMENTARY THEORY OF C��ALGEBRAS
the Gel�fand topology on $�A� is the weakest topology for which all functions �A are continuousIn particular� a basis for this topology is formed by all open sets of the form
�A���O� � f� � $�A�j��A� � Og� �����
where A � A and O is an open set in C Seen as a map from A to C�$�A��� the map A � �A de�ned by ����� is called the Gel fand
transformFor any compact Hausdor� space X � we regard the space C�X� of all continuous functions on
X as a Banach space in the sup�norm de�ned by
k f k��� supx�X
jf�x�j� �����
A basic fact of topology and analysis is that C�X� is complete in this norm Convergence in thesup�norm is the same as uniform convergence What�s more� it is easily veri�ed that C�X� is evena commutative Banach algebra under pointwise addition and multiplication� that is�
��f " �g��x� �� �f�x� " �g�x��
�fg��x� �� f�x�g�x�� �����
Hence the function �X which is � for every x is the unit I One checks that the spectrum off � C�X� is simply the set of values of f
We regard C�$�A�� as a commutative Banach algebra in the manner explained
Theorem ��� Let A be a unital commutative Banach algebra�
�� The Gel�fand transform is a homomorphism from A to C�$�A���
�� The image of A under the Gel�fand transform separates points in $�A��
�� The spectrum of A � A is the set of values of �A on $�A�� in other words
�A� � � �A� � f �A���j� � $�A�g� �����
�� The Gel�fand transform is a contraction that is
k �A k��k A k � �����
The �rst property immediately follows from ����� and ����� When �� � �� there is an A � Afor which ���A� � ���A�� so that �A���� � �A���� This proves ����
If A � G�A� �ie� A is invertibe�� then ��A���A��� � �� so that ��A� � � for all � � $�A�When A �� G�A� the ideal IA �� fABj� B � Ag does not contain I� so that it is contained ina maximal ideal I �this conclusion is actually nontrivial� relying on the axiom of choice in theguise of Hausdor��s maximality priciple� Hence by Theorem ��� there is a � � $�A� for which��A� � � All in all� we have showed that A � G�A� is equivalent to ��A� � � for all � � $�A�Hence A� z � G�A� i� ��A� � z for all � � $�A� Thus the resolvent is
��A� � fz � C j z � ��A��� � $�A�g� �����
Taking the complement� and using ������ we obtain �����Eq ����� then follows from ������ ������ ������ and ����� �
We now look at an example� which is included for three reasons� �rstly it provides a concreteillustration of the Gel�fand transform� secondly it concerns a commutative Banach algebra which isnot a C��algebra� and thirdly the Banach algebra in question has no unit� so the example illustrateswhat happens to the structure theory in the absence of a unit In this connection� let us note ingeneral that each � � $�A� has a suitable extension !� to AI� namely
��A " �I� �� ��A� " �� �����
��� Commutative Banach algebras ��
The point is that !� remains multiplicative on AI� as can be seen from ����� and the de�nition����� This extension is clearly unique Even if one does not actually extend A to AI� the existenceof !� shows that � satis�es ��� �� since this property �which was proved for the unital case� holdsfor !�� and therefore certainly for the restriction � of !� to A
Consider A � L��R�� with the usual linear structure� and norm
k f k���ZR
dx jf�x�j� �����
The associative product � de�ning the Banach algebra structure is convolution� that is�
f � g�x� ��
ZR
dy f�x� y�g�y�� ��� �
Strictly speaking� this should �rst be de�ned on the dense subspace Cc�R�� and subsequently beextended by continuity to L��R�� using the inequality below Indeed� using Fubini�s theorem onproduct integrals� we estimate
k f � g k��ZR
dx jZR
dy f�x� y�g�y�j �ZR
dy jg�y�jZR
dx jf�x� y�j
�
ZR
dy jg�y�jZR
dx jf�x�j �k f k� k g k��
which is ����There is no unit in L��R�� since from ��� � one sees that the unit should be Dirac�s delta�
function �ie� the measure on R which assigns � to x � � and � to all other x�� which does not liein L��R�
We know from the discussion following ����� that every multiplicative functional � � $�L��R��is continuous Standard Banach space theory says that the dual of L��R� is L��R� Hence foreach � � $�L��R�� there is a function �� � L��R� such that
��f� �
ZR
dx f�x����x�� �����
The multiplicativity condition ����� then implies that ���x " y� � ���x����y� for almost allx� y � R This implies
���x� � exp�ipx� �����
for some p � C � and since �� is bounded �being in L��R�� it must be that p � R The functional �corresponding to ����� is simply called p It is clear that di�erent p�s yield di�erent functionals�so that $�L��R�� may be identi�ed with R With this notation� we see from ����� and ����� thatthe Gel�fand transform ����� reads
�f�p� �
ZR
dx f�x�eipx� �����
Hence the Gel�fand transform is nothing but the Fourier transform �more generally� many of theintegral transforms of classical analysis may be seen as special cases of the Gel�fand transform�The well�known fact that the Fourier transform maps the convolution product ��� � into thepointwise product is then a restatement of Theorem ���� Moreover� we see from ���� thatthe spectrum �f� of f in L��R� is just the set of values of its Fourier transform
Note that the Gel�fand transform is strictly a contraction� ie� there is no equality in the bound����� Finally� the Riemann�Lebesgue lemma states that f � L��R� implies �f � C��R�� whichis the space of continuous functions on R that go to zero when jxj � � This is an importantfunction space� whose de�nition may be generalized as follows
Denition ��� Let X be a Hausdor� space X which is locally compact in that each pointhas a compact neighbourhood� The space C��X� consists of all continuous functions on X whichvanish at innity in the sense that for each � � there is a compact subset K � X such thatjf�x�j � � for all x outside K�
�� � ELEMENTARY THEORY OF C��ALGEBRAS
So when X is compact one trivially has C��X� � C�X� When X is not compact� the sup�norm����� can still be de�ned� and just as for C�X� one easily checks that C��X� is a Banach algebrain this norm
We see that in the example A � L��R� the Gel�fand transform takes values in C��$�A�� Thismay be generalized to arbitrary commutative non�unital Banach algebras The non�unital versionof Theorem ��� is
Theorem ��� Let A be a non�unital commutative Banach algebra�
�� The structure space $�A� is locally compact and Hausdor� in the Gel�fand topology�
�� The space $�AI� is the one�point compacti�cation of $�A��
�� The Gel�fand transform is a homomorphism from A to C��$�A���
�� The spectrum of A � A is the set of values of �A on $�A� with zero added if � is not alreadycontained in this set�
�� The claims � and � in Theorem ����� hold�
Recall that the one�point compactication !X of a non�compact topological space X is theset X ��� whose open sets are the open sets in X plus those subsets of X �� whose complementis compact in X If� on the other hand� !X is a compact Hausdor� space� the removal of some point��� yields a locally compact Hausdor� space X � !Xnf�g in the relative topology �ie� the opensets in X are the open sets in !X minus the point ��� whose one�point compacti�cation is� in turn�!X
To prove ��� we add a unit to A� and note that
$�AI� � $�A� ��� �� ��
where each � � $�A� is seen as a functional !� on AI by ������ and the functional � is de�ned by
��A " �I� �� �� �� ��
There can be no other elements � of $�AI�� because the restriction of � has a unique multiplicativeextension ����� to AI� unless it identically vanishes on $�A� In the latter case �� �� is clearlythe only multiplicative possibility
By Proposition ��� the space $�AI� is compact and Hausdor�� by �� �� one has
$�A� � $�AI�nf�g �� ��
as a set In view of the paragraph following ���� in order to prove ���� and �� we need to showthat the Gel�fand topology of $�AI� restricted to $�A� coincides with the Gel�fand topology of$�A� itself Firstly� it is clear from ����� that any open set in $�A� �in its own Gel�fand topology�is the restriction of some open set in $�AI�� because A � AI Secondly� for any A � A� � � C � andopen set O � C � from ����� we evidently have
f� � $�AI�j��A " �I� � Ognf�g � f� � $�A�j��A� � O � �g��When � does not lie in the set f� � �g on the left�hand side� one should here omit the %nf�g&�With ������ this shows that the restriction of any open set in $�AI� to $�A� is always open inthe Gel�fand topology of $�A� This establishes ���� and �
It follows from ����� and �� �� that
�A��� � � �� ��
for all A � A� which by continuity of �A leads to ����The comment preceding Theorem ��� implies ���� The �nal claim follows from the fact
that it holds for AI �
��� Commutative C��algebras ��
��� Commutative C�algebras
The Banach algebra C�X� considered in the previous section is more than a Banach algebra RecallDe�nition ��� The map f � f�� where
f��x� �� f�x�� �� ��
evidently de�nes an involution on C�X�� in which C�X� is a commutative C��algebra with unitThe main goal of this section is to prove the converse statement� cf De�nition ����
Theorem ��� Let A be a commutative C��algebra with unit� Then there is a compact Haus�dor� space X such that A is isometrically isomorphic to C�X�� This space is unique up tohomeomorphism�
The isomorphism in question is the Gel�fand transform� so that X � $�A�� equipped with theGel�fand topology� and the isomorphism � � A� C�X� is given by
��A� �� �A� �� ��
We have already seen in ���� that this transform is a homomorphism� so that ����� is satis�edTo show that ����� holds as well� it su�ces to show that a self�adjoint element of A is mappedinto a real�valued function� because of ������ �� ��� and the fact that the Gel�fand transform iscomplex�linear
We pick A � AR and � � $�A�� and suppose that ��A� � � " i�� where �� � � R By �����one has ��B� � i�� where B �� A� �I is self�adjoint Hence for t � R one computes
j��B " itI�j� � �� " �t� " t�� �� �
On the other hand� using ��� � and ��� � we estimate
j��B " itI�j� �k B " itIk��k �B " itI���B " itI� k�k B� " t� k�k B k� "t��
Using �� � then yields �� " t� �k B k� for all t � R For � � this is impossible For � � �we repeat the argument with B � �B� �nding the same absurdity Hence � � �� so that ��A�is real when A � A� Consequently� by ����� the function �A is real�valued� and ����� follows asannounced
We now prove that the Gel�fand transform� and therefore the morphism � in �� ��� is isometricWhen A � A�� the axiom ��� � reads k A� k�k A k� This implies that k A�m k�k A k�m for allm � N Taking the limit in ����� along the subsequence n � �m then yields
r�A� �k A k � �� ��
In view of ����� and ������ this implies
k �A k��k A k � �� ��
For general A � A we note that A�A is self�adjoint� so that we may use the previous result and��� � to compute
k A k��k A�A k�k dA�A k��k �A� �A k��k �A k�� �
In the third equality we used cA� � �A�� which we just proved� and in the fourth we exploited thefact that C�X� is a C��algebra� so that ��� � is satis�ed in it Hence �� �� holds for all A � A
It follows that � in �� �� is injective� because if ��A� � � for some A � �� then � would failto be an isometry �A commutative Banach algebra for which the Gel�fand transform is injectiveis called semi�simple Thus commutative C��algebraa are semi�simple�
We �nally prove that the morphism � is surjective We know from �� �� that the image
��A� � �A is closed in C�$�A��� because A is closed �being a C��algebra� hence a Banach space�In addition� we know from ���� that ��A� separates points on $�A� Thirdly� since the Gel�fandtransform was just shown to preserve the adjoint� ��A� is closed under complex conjugation by�� �� Finally� since �I � �X by ����� and ������ the image ��A� contains �X The surjectivityof � now follows from the following Stone�Weierstrass theorem� which we state without proof
� � ELEMENTARY THEORY OF C��ALGEBRAS
Lemma ��� Let X be a compact Hausdor� space and regard C�X� as a commutative C��algebra as explained above� A C��subalgebra of C�X� which separates points on X and contains�X coincides with C�X��
Being injective and surjective� the morphism � is bijective� and is therefore an isomorphismThe uniqueness of X is the a consequence of the following result
Proposition ��� Let X be a compact Hausdor� space and regard C�X� as a commutative C��algebra as explained above� Then $�C�X�� equipped with the Gel�fand topology is homeomorphicto X�
Each x � X de�nes a linear map �x � C�X� � C by �x�f� �� f�x�� which is clearly multiplica�tive and nonzero Hence x � �x de�nes a map E �for Evaluation� from X to $�C�X��� givenby
E�x� � f � f�x�� �� ��
Since a compact Hausdor� space is normal� Urysohn�s lemma says that C�X� separates points onX �ie� for all x � y there is an f � C�X� for which f�x� � f�y�� This shows that E is injective
We now use the compactness of X and Theorem ��� to prove that E is surjective Themaximal ideal Ix �� I�x in C�X� which corresponds to �x � $�C�X�� is obviously
Ix � ff � C�X�j f�x� � �g� �����
Therefore� when E is not surjective there exists a maximal ideal I � C�X� which for each x �X contains at a function fx for which fx�x� � � �if not� I would contain an ideal Ix whichthereby would not be maximal� For each x� the set Ox where fx is nonzero is open� becausef is continuous This gives a covering fOxgx�X of X By compactness� there exists a �nite
subcovering fOxigi�������N Then form the function g ��PN
i�� jfxi j� This function is strictlypositive by construction� so that it is invertible �note that f � C�X� is invertible i� f�x� � � forall x � X � in which case f���x� � ��f�x�� But I is an ideal� so that� with all fxi � I �since allfx � I� also g � I But an ideal containing an invertible element must coincide with A �see thecomment after ����� contradicting the assumption that I is a maximal ideal
Hence E is surjective� since we already found it is injective� E must be a bijection It remainsto be shown that E is a homeomorphism Let Xo denote X with its originally given topology� andwrite XG for X with the topology induced by E�� Since �f �E � f by �� �� and ������ and the
Gel�fand topology on $�C�X�� is the weakest topology for which all functions �f are continuous�we infer that XG is weaker than Xo �since f � lying in C�Xo�� is continuous� Here a topology T�is called weaker than a topology T� on the same set if any open set of T� contains an open set ofT� This includes the possibility T� � T�
Without proof we now state a result from topology
Lemma ��� Let a set X be Hausdor� in some topology T� and compact in a topology T�� If T�is weaker than T� then T� � T��
Since Xo and XG are both compact and Hausdor� �the former by assumption� and the lat�ter by Proposition ����� we conclude from this lemma that X� � XG� in other words� E is ahomeomorphism This concludes the proof of ��� �
Proposition ��� shows that X as a topological space may be extracted from the Banach�algebraic structure of C�X�� up to homeomorphism Hence if C�X� � C�Y � as a C��algebra�where Y is a second compact Hausdor� space� then X � Y as topological spaces Given theisomorphism A � C�X� constructed above� a second isomorphism A � C�Y � is therefore onlypossible if X � Y This proves the �nal claim of Theorem ��� �
The condition that a compact topological space be Hausdor� is su�cient� but not necessaryfor the completeness of C�X� in the sup�norm However� when X is not Hausdor� yet C�X� iscomplete� the map E may fail to be injective since in that case C�X� may fail to separate pointson X
��� Commutative C��algebras ��
On the other hand� suppose X is locally compact but not compact� and consider A � Cb�X��this is the space of all continuous bounded functions on X Equipped with the operations ������������ and �� �� this is a commutative C��algebra The map E � X � $�Cb�X�� is now injective�but fails to be surjective �this is suggested by the invalidity of the proof we gave for C�X�� Indeed�it can be shown that $�Cb�X�� is homeomorphic to the Ceh�Stone compacti�cation of X
Let us now consider what happens to Theorem ��� when A has no unit Following the strategywe used in proving Theorem ���� we would like to add a unit to A As in the case of a generalBanach algebra �cf section ���� we form AI by ������ de�ne multiplication by ������ and use thenatural involution
�A " �I�� �� A� " �I� �����
However� the straightforward norm ����� cannot be used� since it is not a C��norm in that axiom��� � is not satis�ed Recall De�nition ���
Lemma ��� Let A be a C��algebra�
�� The map � � A� B�A� given by��A�B �� AB �����
establishes an isomorphism between A and ��A� � B�A��
�� When A has no unit de�ne a norm on AI by
k A " �Ik��k ��A� " �Ik� �����
where the norm on the right�hand side is the operator norm ��� in B�A� and I on theright�hand side is the unit operator in B�A�� With the operations ���� and ���� thenorm ���� turns AI into a C��algebra with unit�
By ����� we have k ��A�B k�k AB k�k A kk B k for all B� so that k ��A� k�k A k by ����On the other hand� using ��� � and ����� we can write
k A k�k AA� k � k A k�k ��A�A�
k A k k� ��A� k�
in the last step we used ���� and k �A�� k A k� k� � Hence
k ��A� k�k A k � �����
Being isometric� the map � must be injective� it is clearly a homomorphism� so that we have proved����
It is clear from ����� and ����� that the map A" �I� ��A� " �I �where the symbol I on theleft�hand side is de�ned below ������ and the I on the right�hand side is the unit in B�A�� is amorphism Hence the norm ����� satis�es ������ because ���� is satis�ed in B�A� Moreover� inorder to prove that the norm ����� satis�es ��� �� by Lemma ���� it su�ces to prove that
k ��A� " �Ik��k ���A� " �I�����A� " �I� k �����
for all A � A and � � C To do so� we use a trick similar to the one involving ������ but with infreplaced by sup Namely� in view of ����� for given A � B�B� and � � there exists a v � V � withk v k� �� such that k A k� �� �k Av k� Applying this with B � A and A� ��A� " �I� we inferthat for every � � there exists a B � A with norm � such that
k ��A� " �Ik� �� �k ���A� " �I�B k��k AB " �B k��k �AB " �B���AB " �B� k �Here we used ��� � in A Using ������ the right�hand side may be rearranged as
k ��B����A� " �I���A" �I�B k�k ��B�� k k ���A� " �I�����A� " �I� k k B k �Since k ��B�� k�k B� k�k B k� � by ����� and ������ and k B k� � also in the last term� theinequality ����� follows by letting �� � �
Hence the C��algebraic version of Theorem ��� is
�� � ELEMENTARY THEORY OF C��ALGEBRAS
Proposition ��� For every C��algebra without unit there exists a unique unital C��algebra AIand an isometric hence injective morphism A� AI such that AI�A � C �
The uniqueness of AI follows from Corollary ��� below On the other hand� in view of thefact that both ����� and ����� de�ne a norm on AI satisfying the claims of Proposition ���� weconclude that the unital Banach algebra AI called for in that proposition is not� in general� unique
In any case� having established the existence of the unitization of an arbitrary non�unital C��algebra� we see that� in particular� a commutative non�unital C��algebra has a unitization Thepassage from Theorem ��� to Theorem ��� may then be repeated in the C��algebraic setting�the only nontrivial point compared to the situation for Banach algebras is the generalization ofLemma ��� This now reads
Lemma ��� Let X be a locally compact Hausdor� space and regard C��X� as a commutativeC��algebra as explained below De�nition ������
A C��subalgebra A of C��X� which separates points on X and is such that for each x � Xthere is an f � A such that f�x� � � coincides with C��X��
At the end of the day we then �nd
Theorem ��� Let A be a commutative C��algebra without unit� There is a locally compactHausdor� space X such that A is isometrically isomorphic to C��X�� This space is unique up tohomeomorphism�
��� Spectrum and functional calculus
We return to the general case in which a C��algebra A is not necessarily commutative �but assumedunital�� but analyze properties of A by studying certain commutative subalgebras This will leadto important results
For each element A � A there is a smallest C��subalgebra C��A� I� of A which contains Aand I� namely the closure of the linear span of I and all operators of the type A� � � � An� whereAi is A or A� Following the terminology for operators on a Hilbert space� an element A � A iscalled normal when �A�A� � � The crucial property of a normal operator is that C��A� I� iscommutative In particular� when A is self�adjoint� C��A� I� is simply the closure of the space ofall polynomials in A It is su�cient for our purposes to restrict ourselves to this case
Theorem ��� Let A � A� be a self�adjoint element of a unital C��algebra�
�� The spectrum A�A� of A in A coincides with the spectrum C��A�I�A� of A in C��A� I� sothat we may unambiguously speak of the spectrum �A��
�� The spectrum �A� is a subset of R�
�� The structure space $�C��A� I�� is homeomorphic with �A� so that C��A� I� is isomorphicto C� �A��� Under this isomorphism the Gel�fand transform �A � �A� � R is the identityfunction id��A � t� t�
Recall ����� Let A � G�A� be normal in A� and consider the C��algebra C��A�A��� I�generated by A� A��� and I One has �A���� � �A����� and A� A�� A��� �A���� and Iall commutewith each other Hence C��A�A��� I� is commutative� it is the closure of the space of all polynomialsin A� A�� A��� �A����� and I By Theorem ��� we have C��A�A��� I�� C�X� for some compactHausdor� space X Since A is invertible and the Gel�fand transform ����� is an isomorphism� �Ais invertible in C�X� �ie� �A�x� � �x for all x � X� However� for any f � C�X� that is nonzerothroughout X we have � �k f k��� ff� � � pointwise� so that � � �X� k f k��� ff� � � pointwise�hence
k �X � ff � � k f k��k�� ��
Here f� is given by �� �� Using Lemma ���� in terms of I� �X we may therefore write
�
f�
f�
k f k��
�Xk��
�I� ff�
k f k��
�k� ��� �
��� Spectrum and functional calculus ��
Hence �A�� is a norm�convergent limit of a sequence of polynomials in �A and �A� Gel�fand trans�forming this result back to C��A�A��� I�� we infer that A�� is a norm�convergent limit of a sequenceof polynomials in A and A� Hence A�� lies in C��A� I�� and C��A�A��� I� � C��A� I�
Now replace A by A�z� where z � C When A is normal A�z is normal So if we assume thatA � z � G�A� the argument above applies� leading to the conclusion that the resolvent �A�A� inA coincides with the resolvent �C��A�I�A� in C��A� I� By De�nition ��� we then conclude that A�A� � C��A�I�A�
According to Theorem ���� the function �A is real�valued when A � A� Hence by ���� thespectrum C��A�I�A� is real� so that by the previous result �A� is real
Finally� given the isomorphism C��A� I� � C�X� of Theorem ��� �where X � $�C��A� I����according to ���� the function �A is a surjective map from X to �A� We now prove injectivityWhen ��� �� � X and ���A� � ���A�� then� for all n � N� we have
���An� � ���A�n � ���A�n � ���A
n�
by iterating ����� with B � A Since also ���I� � ���I� � � by ������ we conclude by linearitythat �� � �� on all polynomials in A By continuity �cf ����� this implies that �� � �� onC��A� I�� since the linear span of all polynomials is dense in C��A� I� Using ������ we have provedthat �A���� � �A���� implies �� � ��
Since �A � C�X� by ����� �A is continuous To prove continuity of the inverse� one checksthat for z � �A� the functional �A���z� � $�C��A� I�� maps A to z �and hence An to zn� etc�Looking at ������ one then sees that �A�� is continuous In conclusion� �A is a homeomorphismThe �nal claim in ���� is then obvious �
An immediate consequence of this theorem is the continuous functional calculus
Corollary ��� For each self�adjoint element A � A and each f � C� �A�� there is an operatorf�A� � A which is the obvious expression when f is a polynomial and in general is given via theuniform approximation of f by polynomials such that
�f�A�� � f� �A��� �����
k f�A� k � k f k� � �����
In particular the norm of f�A� in C��A� I� coincides with its norm in A�
Theorem ���� yields an isomorphism C� �A�� � C��A� I�� which is precisely the map f �f�A� of the continuous functional calculus The fact that this isomorphism is isometric �see ����yields ����� Since f� �A�� is the set of values of f on �A�� ����� follows from ������ withA� f�A�
The last claim follows by combining ���� with ����� and ����� �
Corollary ��� The norm in a C��algebra is unique that is given a C��algebra A there is noother norm in which A is a C��algebra�
First assume A � A�� and apply ����� with f � id��A By de�nition �cf ������� the sup�normof id��A is r�A�� so that
k A k� r�A� �A � A��� �����
Since A�A is self�adjoint for any A� for general A � A we have� using ��� ��
k A k�pr�A�A�� �����
Since the spectrum is determined by the algebraic structure alone� ����� shows that the norm isdetermined by the algebraic structure as well �
Note that Corollary ��� does not imply that a given ��algebra can be normed only in one wayso as to be completed into a C��algebra �we will� in fact� encounter an example of the oppositesituation� In ��� the completeness of A is assumed from the outset
�� � ELEMENTARY THEORY OF C��ALGEBRAS
Corollary ��� A morphism � � A� B between two C��algebras satis�es
k ��A� k�k A k� �����
and is therefore automatically continuous�
When z � ��A�� so that �A � z��� exists in A� then ��A � z� is certainly invertible in B� for����� implies that ���A � z���� � ���A � z���� Hence ��A� ����A��� so that
���A�� �A�� �����
Hence r���A�� � r�A�� so that ����� follows from ����� �
For later use we note
Lemma ��� When � � A� B is a morphism and A � A� then
f���A�� � ��f�A�� �����
for all f � C� �A�� here f�A� is de�ned by the continuous functional calculus and so is f���A��in view of �����
The property is true for polynomials by ������ since for those f has its naive meaning Forgeneral f the result then follows by continuity �
�� Positivity in C�algebras
A bounded operator A � B�H� on a Hilbert space H is called positive when ��� A�� � � forall � � H� this property is equivalent to A� � A and �A� R� � and clearly also applies toclosed subalgebras of B�H� In quantum mechanics this means that the expectation value of theobservable A is always positive
Classically� a function f on some space X is positive simply when f�x� � � for all x � X Thisapplies� in particular� to elements of the commutative C��algebra C��X� �where X is a locallycompact Hausdor� space� Hence we have a notion of positivity for certain concrete C��algebras�which we would like to generalize to arbitrary abstract C��algebras Positivity is one of the mostimportant features in a C��algebra� it will� for example� play a central role in the proof of theGel�fand Neumark theorem In particular� one is interested in �nding a number of equivalentcharacterizations of positivity
Denition ��� An element A of a C��algebra A is called positive when A � A� and its spec�trum is positive� i�e� �A� � R� � We write A � � or A � A� where
A� �� fA � ARj �A� � R�g� �����
It is immediate from Theorems ���� and ���� that A � AR is positive i� its Gel�fandtransform �A is pointwise positive in C� �A��
Proposition ��� The set A� of all positive elements of a C��algebra A is a convex cone� thatis
�� when A � A� and t � R� then tA � A��
�� when A�B � A� then A " B � A��
�� A� � �A� � ��
��� Positivity in C��algebras ��
The �rst property follows from �tA� � t �A�� which is a special case of �����Since �A� ��� r�A� � we have jc � tj � c for all t � �A� and all c � r�A� Hence
supt���A jc���A� �Aj � c by ���� and ����� so that k c���A� �A k�� c Gel�fand transform�ing back to C��A� I�� this implies k cI�A k� c for all c �k A k by ��� Inverting this argument�one sees that if k cI�A k� c for some c �k A k� then �A� � R�
Use this with A� A " B and c �k A k " k B k� clearly c �k A " B k by ���� Then
k cI� �A " B� k�k �k A k �A� k " k �k B k �B� k� c�
where in the last step we used the previous paragraph for A and for B separately As we haveseen� this inequality implies A " B � A�
Finally� when A � A� and A � �A� it must be that �A� � �� hence A � � by ����� and����� �
This is important� because a convex cone in a real vector space is equivalent to a linear partialordering� ie� a partial ordering � in which A � B implies A"C � B"C for all C and �A � �Bfor all � � R� The real vector space in question is the space AR of all self�adjoint elements of AThe equivalence between these two structures is as follows� given A�
R�� A� one de�nes A � B if
B �A � A�R
� and given � one puts A�R
� fA � AR j � � AgFor example� when A � A� one checks the validity of
� k A k I� A �k A k I �����
by taking the Gel�fand transform of C��A� I� The implication
�B � A � B �� k A k�k B k ��� �
then follows� because �B � A � B and ����� for A � B yield � k B k I� A �k B k I� so that �A� �� k B k� k B k � hence k A k�k B k by ����� and ����� For later use we also record
Lemma ��� When A�B � A� and k A " B k� k then k A k� k�
By ����� we have A " B � kI� hence � � A � kI�B by the linearity of the partial ordering�which also implies that kI� B � kI� as � � B Hence� using �kI� � �since k � �� we obtain�kI� A � kI� from which the lemma follows by ��� � �
We now come to the central result in the theory of positivity in C��algebras� which generalizesthe cases A � B�H� and A � C��X�
Theorem ��� One has
A� � fA�jA � ARg �����
� fB�BjB � Ag� �����
When �A� � R� and A � A� thenpA � AR is de�ned by the continuous functional calculus
for f �p�� and satis�es
pA�
� A Hence A� fA�jA � ARg The opposite inclusion followsfrom ����� and ���� This proves �����
The inclusion A� fB�BjB � Ag is is trivial from �����
Lemma ��� Every self�adjoint element A has a decomposition A � A� �A� where A�� A� �A� and A�A� � �� Moreover k A k�k A k�
Apply the continuous functional calculus with f � id��A � f� � f�� where id��A�t�� f��t� �maxft� �g� and f��t� � maxf�t� �g Since k f k�� r�A� �k A k �where we used ������� thebound follows from ����� with A� A �
We use this lemma to prove that fB�BjB � Ag A� Apply the lemma to A � B�B �notingthat A � A�� Then
�A��� � �A��A� �A��A� � �A�AA� � �A�B�BA� � ��BA���BA��
Since �A�� � R� because A� is positive� we see from ����� with f�t� � t� that �A��� � �Hence ��BA���BA� � �
�� � ELEMENTARY THEORY OF C��ALGEBRAS
Lemma ��� If �C�C � A� for some C � A then C � ��
By ����� we can write C � D " iE� D�E � AR� so that
C�C � �D� " �E� � CC�� �����
Now for any A�B � A one has
�AB� � f�g � �BA� � f�g� �����
This is because for z � � the invertibility of AB�z implies the invertibility of BA�z Namely� onecomputes that �BA� z��� � B�AB � z���A� z��I Applying this with A� C and B � C� wesee that the assumption �C�C� � R� implies �CC�� � R� � hence ��CC�� � R� By ������������ and � �� we see that C�C � �� ie� �C�C� � R� � so that the assumption �C�C � A�now yields �C�C� � � Hence C � � by � �� �
The last claim before the lemma therefore implies BA� � � As �A��� � ��BA���BA� � �we see that �A��� � �� and �nally A� � � by the continuous functional calculus with f�t� � t���Hence B�B � A�� which lies in A� �
An important consequence of ����� is the fact that inequalities of the type A� � A� forA�� A� � AR are stable under conjugation by arbitrary elements B � A� so that A� � A� impliesB�A�B � B�A�B This is because A� � A� is the same as A��A� � �� by ����� there is an A� � Asuch that A��A� � A��A� But clearly �A�B��A�B � �� and this is nothing but B�AB � B�A�BFor example� replace A in ����� by A�A� and use ��� �� yielding A�A �k A k� I Applying theabove principle gives
B�A�AB �k A k� B�B �����
for all A�B � A
��� Ideals in C�algebras
An ideal I in a C��algebra A is de�ned by ��� As we have seen� a proper ideal cannot containI� in order to prove properties of ideals we need a suitable replacement of a unit
Denition ��� An approximate unit in a non�unital C��algebra A is a family fI�g��� where� is some directed set i�e� a set with a partial order and a sense in which � � � with thefollowing properties�
��
I�� � I� �����
and �I�� � ��� � so that
k I� k� �� �����
��
lim���
k I�A�A k� lim���
k AI��A k� � �����
for all A � A�For example� the C��algebra C��R� has no unit �the unit would be �R� which does not vanish
at in�nity because it is constant�� but an approximate unit may be constructed as follows� take� � N� and take In to be a continuous function which is � on ��n� n and vanishes for jxj n" �One checks the axioms� and notes that one certainly does not have In� �R in the sup�norm
Proposition ��� Every non�unital C��algebra A has an approximate unit� When A is separablein containing a countable dense subset then � may be taken to be countable�
�� Ideals in C��algebras ��
One takes � to be the set of all �nite subsets of A� partially ordered by inclusion Hence � � �is of the form � � fA�� � � � � Ang� from which we build the element B� ��
Pi A
�iAi Clearly B� is
self�adjoint� and according to � � and � �� one has �B� � R� � so that n��I"B� is invertiblein AI Hence we may form
I� �� B��n��I" B����� �����
Since B� is self�adjoint and B� commutes with functions of itself �such as �n��I"B������ one has
I�� � I� Although �n��I"B���� is computed in AI� so that it is of the form C"�I for some C � A
and � � C � one has I� � B�C "�B�� which lies in A Using the continuous functional calculus onB� with f�t� � t��n " t�� one sees from ����� and the positivity of B� that �I�� � ��� �
Putting Ci �� I�Ai �Ai� a simple computation shows thatXi
CiC�i � n��B��n��I" B����� ��� �
We now apply ����� with A � B� and f�t� � n��t�n�� " t��� Since f � � and f assumes itsmaximum at t � ��n� one has supt�R� jf�t�j � ���n As �B� � R� � it follows that k f k�� ���n�hence k n��B��n��I" B���� k� ���n by ������ so that kPi CiC
�i k� ���n by ��� � Lemma
� � then shows that k CiC�i k� ���n for each i � �� � � � � n Since any A � A sits in some directedsubset of � with n��� it follows from ��� � that
lim���
k I�A�A k�� lim���
k �I�A�A��I�A�A k� lim���
k C�i Ci k� ��
The other equality in ����� follows analogouslyFinally� when A is separable one may draw all Ai occurring as elements of � � � from a
countable dense subset� so that � is countable �
The main properties of ideals in C��algebras are as follows
Theorem ��� Let I be an ideal in a C��algebra A�
�� If A � I then A� � I� in other words every ideal in a C��algebra is self�adjoint�
�� The quotient A�I is a C��algebra in the norm ���� the multiplication ���� and theinvolution
��A�� �� ��A��� �����
Note that ����� is well de�ned because of ����Put I� �� fA�jA � Ig Note that J � I implies J�J � I� I�� it lies in I because I is an ideal�
hence a left�ideal� and it lies in I� because I� is an ideal� hence a right�ideal Since I is an ideal�I�I� is a C��subalgebra of A Hence by ��� it has an approximate unit fI�g Take J � I Using��� � and ������ we estimate
k J� � J�I� k��k �J � I�J��J� � J�I�� k�k �JJ� � JJ�I��� I��JJ� � JJ�I�� k
�k �JJ� � JJ�I�� k " k I��JJ� � JJ�I�� k�k �J�J � J�JI�� k " k I� k k �JJ� � JJ�I�� k �As we have seen� J�J � I � I�� so that� also using ������ both terms vanish for � � � Hencelim��� k J��J�I� k� � But I� lies in I� I�� so certainly I� � I� and since I is an ideal it mustbe that J�I� � I for all � Hence J� is a norm�limit of elements in I� since I is closed� it followsthat J� � I This proves ����
In view of ����� all we need to prove to establish ���� is the property ��� � This uses
Lemma ��� Let fI�g be an approximate unit in I and let A � A� Then
k ��A� k� lim���
k A�AI� k � �����
�� � ELEMENTARY THEORY OF C��ALGEBRAS
It is obvious from ����� that
k A�AI� k�k ��A� k � �����
To derive the opposite inequality� add a unit I to A if necessary� pick any J � I� and write
k A� AI� k�k �A " J��I� I�� " J�I�� Ik�k A " J k k I� I� k " k JI�� J k �Note that
k I� I� k� � ������
by ���� and the proof of � � The second term on the right�hand side goes to zero for ����since J � I Hence
lim���
k A�AI� k�k A " J k � ������
For each � � we can choose J � I so that ����� holds For this speci�c J we combine ������������� and ����� to �nd
lim���
k A�AI� k �� �k ��A� k�k A�AI� k �
Letting �� � proves ����� �
We now prove ��� � in A�I Successively using ������ ��� � in AI� ������� ������ ������ and������ we �nd
k ��A� k�� lim���
k A�AI� k�� lim���
k �A�AI����A�AI�� k
� lim���
k �I� I��A�A�I� I�� k� lim���
k I� I� k k A�A�I� I�� k� lim���
k A�A�I� I�� k�k ��A�A� k�k ��A���A�� k�k ��A���A�� k �
Lemma ���� then implies ��� � �
This seemingly technical result is very important
Corollary ��� The kernel of a morphism between two C��algebras is an ideal� Conversely everyideal in a C��algebra is the kernel of some morphism� Hence every morphism has norm ��
The �rst claim is almost trivial� since ��A� � � implies ��AB� � ��BA� � � for all B by����� Also� since � is continuous �see ���� its kernel is closed
The converse follows from Theorem ���� since I is the kernel of the canonical projection� � A� A�I� where A�I is a C��algebra� and � is a morphism by ������ and �����
The �nal claim follows from the preceding one� since k � k� � �
For the next consequence of ��� we need a
Lemma ��� An injective morphism between C��algebras is isometric� In particular its rangeis closed�
Assume there is an B � A for which k ��B� k �k B k By ��� �� ������ and ����� thisimplies k ��B�B� k �k B�B k Put A �� B�B� noting that A� � A By ����� and ����� wemust have �A� � ���A�� Then ����� implies ���A�� � �A� By Urysohn�s lemma there is anonzero f � C� �A�� which vanishes on ���A��� so that f���A�� � � By Lemma ��� we have��f�A�� � �� contradicting the injectivity of � �
Corollary ��� The image of a morphism � � A � B between two C��algebras is closed� Inparticular ��A� is a C��subalgebra of B�
De�ne � � A� ker��� � B by ���A � � ��A�� where �A is the equivalence class in A� ker���of A � A By the theory of vector spaces� � is a vector space isomorphism between A� ker���and ��A�� and � � � � � In particular� � is injective According to ��� and ����� the spaceA� ker��� is a C��algebra Since � and � are morphisms� � is a C��algebra morphism Hence��A� ker���� has closed range in B by �� But ��A� ker���� � ��A�� so that � has closed rangein B Since � is a morphism� its image is a ��algebra in B� which by the preceding sentence isclosed in the norm of B Hence ��A�� inheriting all operations in B� is a C��algebra �
�� States ��
��� States
The notion of a state in the sense de�ned below comes from quantum mechanics� but states playa central role in the general theory of abstract C��algebras also
Denition ��� A state on a unital C��algebra A is a linear map � � A� C which is positivein that ��A� � � for all A � A� and normalized in that
��I� � �� ������
The state space S�A� of A consists of all states on A�
For example� when A B�H� then every � � H with norm � de�nes a state � by
��A� �� ��� A��� ������
Positivity follows from Theorem � �� since ��B�B� �k B� k�� �� and normalization is obviousfrom ��I� � ����� � �
Theorem ��� The state space of A � C�X� consists of all probability measures on X�
By the Riesz theorem of measure theory� each positive linear map � � C�X� � C is given by aregular positive measure �� on X The normalization of � implies that ���X� � ���X� � �� sothat �� is a probability measure �
The positivity of � with � � implies that �A�B�� �� ��A�B� de�nes a pre�inner product onA Hence from ���� we obtain
j��A�B�j� � ��A�A���B�B�� ������
which will often be used Moreover� for all A � A one has
��A�� � ��A�� ������
as ��A�� � ��A�I� � �A� I�� � �I� A�� � ��A�Partly in order to extend the de�nition of a state to non�unital C��algebras� we have
Proposition ��� A linear map � � A � C on a unital C��algebra is positive i� � is boundedand
k � k� ��I�� ���� �
In particular�
�� A state on a unital C��algebra is bounded with norm ��
�� An element � � A� for which k � k� ��I� � � is a state on A�
When � is positive and A � A� we have� using ������ the bound j��A�j � ��I� k A k Forgeneral A we use ������ with A � I� ��� �� and the bound just derived to �nd
j��B�j� � ��B�B���I�� ��I�� k B�B k� ��I�� k B k� �Hence k � k� ��I� Since the upper bound is reached by B � I� we have ���� �
To prove the converse claim� we �rst note that the argument around �� � may be copied�showing that � is real on AR Next� we show that A � � implies ��A� � � Choose s � smallenough� so that k I� sA k� � Then �assuming � � ��
� �k I� sA k� k � k��I�
k I� sA k� j��I� sA�j��I�
�
Hence j��I�� s��A�j � ��I�� which is only possible when ��A� � � �
We now pass to states on C��algebras without unit Firstly� we look at a state in a more generalcontext
� � ELEMENTARY THEORY OF C��ALGEBRAS
Denition ��� A positive map Q � A� B between two C��algebras is a linear map with theproperty that A � � implies Q�A� � � in B�
Proposition ��� A positive map between two C��algebras is bounded continuous�
Let us �rst show that boundedness on A� implies boundedness on A Using ����� and � ��we can write
A � A�� �A�� " iA��� � iA���� ������
where A�� etc are positive Since k A� k�k A k and k A�� k�k A k by ������ we have k B k�k A kfor B � A��� A
��� A
���� or A��� by � � Hence if k Q�B� k� C k B k for all B � A� and some
C �� then
k Q�A� k�k Q�A��� k " k Q�A��� k " k Q�A���� k " k Q�A���� k� �C k A k �
Now assume that Q is not bounded� by the previous argument it is not bounded on A�� so thatfor each n � N there is an An � A�� so that k Q�An� k� n� �here A�� consists of all A � A� withk A k� �� The series
P�n�� n
��An obviously converges to some A � A� Since Q is positive� wehave Q�A� � n��Q�An� � � for each n Hence by ��� �
k Q�A� k� n�� k Q�An� k� n
for all n � N� which is impossible since k Q�A� k is some �nite number Thus Q is bounded onA�� and therefore on A by the previous paragraph �
Choosing B � C � we see that a state on a unital C��algebra is a special case of a positive mapbetween C��algebras� Proposition ��� then provides an alternative proof of ���� Hence in thenon�unital case we may replace the normalization condition in ��� as follows
Denition ��� A state on a C��algebra A is a linear map � � A� C which is positive and hasnorm ��
This de�nition is possible by ���� and is consistent with ��� because of ���� � The followingresult is very useful� cf ��
Proposition ��� A state � on a C��algebra without unit has a unique extension to a state �Ion the unitization AI�
The extension in question is de�ned by
�I�A " �I� �� ��A� " �� ������
This obviously satis�es ������� it remains to prove positivity
Since a state � on A is bounded by ���� we have j��A�AI��j � � for any approximate unitin A The derivation of ������ and ������ may then be copied from the unital case� in particular�one still has j��A�j� � ��A�A� Combining this with ������� we obtain from ������ that
�I��A " �I���A " �I��� j��A� " �j� � ��
Hence � is positive by ����� �
There are lots of states�
Lemma ��� For every A � A and a � �A� there is a state �a on A for which ��A� � a� WhenA � A� there exists a state � such that j��A�j �k A k�
��� Representations and the GNS�construction ��
If necessary we add a unit to A �this is justi�ed by ���� De�ne a linear map !�a � CA"C I� C
by !�a��A " �I� �� �a " � Since a � �A� one has �a " � � ��A " �I�� this easily follows fromthe de�nition of Hence ����� with A� �A " �I implies j!�a��A " �I�j �k ��A " �I� k Since!�a�I� � �� it follows that k !� k� � By the Hahn�Banach Theorem �� � there exists an extension�a of !� to A of norm � By ���� �a is a state� which clearly satis�es �a�A� � !�a�A� � a
Since �A� is closed by ����� there is an a � �A� for which r�A� � jaj For this a one hasj��A�j � jaj � r�A� �k A k by ����� �
An important feature of a state space S�A� is that it is a convex set A convex set C in avector space C is a subset of V such that the convex sum �v " �� � ��w belongs to C wheneverv� w � C and � � ��� � Repeating this process� it follows that
Pi pivi belongs to C when all
pi � � andP
i pi � �� and all vi � C In the unital case it is clear that S�A� is convex� since bothpositivity and normalization are clearly preserved under convex sums In the non�unital case onearrives at this conclusion most simply via ���
We return to the unital case Let S�A� be the state space of a unital C��algebra A We sawin ��� that each element � of S�A� is continuous� so that S�A� � A� Since w��limits obviouslypreserve positivity and normalization� we see that S�A� is closed in A� if the latter is equippedwith the w��topology Moreover� S�A� is a closed subset of the unit ball of A� by ����� so thatS�A� is compact in the �relative� w��topology by the Banach�Alaoglu theorem
It follows that the state space of a unital C��algebra is a compact convex set The verysimplest example is A � C � in which case S�A� is a point
The next case is A � C � C � C � The dual is C � as well� so that each element of �C � �� isof the form ��� '"�� � c��� " c��� Positive elements of C � C are fo the form � '"� with � � �and � � �� so that a positive functional must have c� � � and c� � � Finally� since I � � '"��normalization yields c� " c� � � We conclude that S�C � C � may be identi�ed with the interval��� �
Now consider A � M��C � We identify M��C � with its dual through the pairing ��A� �Tr�A It follows that S�A� consists of all positive � � � matrices � with Tr � � �� these are thedensity matrices of quantum mechanics To identify S�A� with a familiar compact convex set� weparametrize
� � �
�
�� " x y " izy � iz �� x
�� ������
where x� y� z � R The positivity of this matrix then corresponds to the constraint x�"y�"z� � �Hence S�M��C �� is the unit ball in R�
�� Representations and the GNSconstruction
The material of this section explains how the usual Hilbert space framework of quantum mechanicsemerges from the C��algebraic setting
Denition ��� A representation of A on a Hilbert space H is a complex linear map � �A� B�H� satisfying
��A � B� � ��A���B��
��A�� � ��A�� ������
for all A�B � A�A representation � is automatically continuous� satisfying the bound
k ��A� k�k A k � ������
This is because � is a morphism� cf ����� In particular� k ��A� k�k A k when � is faithful byLemma ��
There is a natural equivalence relation in the set of all representations of A� two representa�tions ��� �� on Hilbert spaces H��H�� respectively� are called equivalent if there exists a unitaryisomorphism U � H� � H� such that U���A�U� � ���A� for all A � A
�� � ELEMENTARY THEORY OF C��ALGEBRAS
The map ��A� � � for all A � A is a representation� more generally� such trivial � may occuras a summand To exclude this possibility� one says that a representation is non�degenerate if �is the only vector annihilated by all representatives of A
A representation � is called cyclic if its carrier space H contains a cyclic vector ( for �� thismeans that the closure of ��A�( �which in any case is a closed subspace of H� coincides with H
Proposition ��� Any non�degenerate representation � is a direct sum of cyclic representations�
The proof uses a lemma which appears in many other proofs as well
Lemma ��� Let M be a ��algebra in B�H� take a nonzero vector � � H and let p be theprojection onto the closure of M�� Then p �M� that is �p�A � � for all A �M�
If A � M then ApH pH by de�nition of p Hence p�Ap � � with p� � I� p� this readsAp � pAp When A � A� then
�Ap�� � pA � �pAp�� � pAp � Ap�
so that �A� p � � By ����� this is true for all A �M �
Apply this lemma with M � ��A�� the assumption of non�degeneracy guarantees that p isnonzero� and the conclusion implies that A� p��A� de�nes a subrepresentation of A on pH Thissubrepresentation is clearly cyclic� with cyclic vector � This process may be repeated on p�H�etc �
If � is a non�degenerate representation of A on H� then any unit vector � � H de�nes a state� � S�A�� referred to as a vector state relative to �� by means of ������ Conversely� from anystate � � S�A� one can construct a cyclic representation �� on a Hilbert space H� with cyclicvector (� in the following way We restrict ourselves to the unital case� the general case followsby adding a unit to A and using ���
Construction ��� �� Given � � S�A� de�ne the sesquilinear form � � �� on A by
�A�B�� �� ��A�B�� ������
Since � is a state hence a positive functional this form is positive semi�de�nite this meansthat �A�A�� � � for all A� Its null space
N� � fA � A j��A�A� � �g ������
is a closed left�ideal in A�
�� The form � � �� projects to an inner product � � �� on the quotient A�N�� If V � A � A�N�
is the canonical projection then by de�nition
�V A� V B�� �� �A�B��� ������
The Hilbert space H� is the closure of A�N� in this inner product�
�� The representation ���A� is �rstly de�ned on A�N� � H� by
���A�V B �� V AB� ������
it follows that �� is continuous� Hence ���A� may be de�ned on all of H� by continuousextension of ����� where it satis�es ������
�� The cyclic vector is de�ned by (� � V I so that
�(�� ���A�(�� � ��A� �A � A� ���� �
���� The Gel fand�Neumark theorem ��
We now prove the various claims made here First note that the null space N� of � � �� can bede�ned in two equivalent ways�
N� �� fA � A j �A�A�� � �g � fA � A j �A�B�� � � �B � Ag� ������
The equivalence follows from the Cauchy�Schwarz inequality ������ The equality ������ impliesthat N� is a left�ideal� which is closed because of the continuity of � This is important� becauseit implies that the map ��A� � A � A de�ned in ����� quotients well to a map from A�N� toA�N�� the latter map is �� de�ned in ������ Since � is a morphism� it is easily checked that ��is a morphism as well� satisfying ������ on the dense subspace A�N� of H�
To prove that �� is continuous on A�N�� we compute k ���A�� k� for � � V B� whereA�B � A By ������ and step � above� one has k ���A�� k�� ��B�AA � B� By ����� andthe positivity of � one has ��B�AA � B� �k A k� ��B�B� But ��B�B� �k � k�� so thatk ���A�� k�k A k k � k� upon which
k ���A� k�k A k ������
follows from ����
For later use we mention that the GNS�construction yields
����A�(� � ���B�(�� � ��A�B�� ������
Putting B � A yields
k ���A�(� k�� ��A�A�� ������
which may alternatively be derived from ���� � and the fact that �� is a representation
Proposition ��� If ���A��H� is cyclic then the GNS�representation ����A��H�� de�ned by anyvector state ( corresponding to a cyclic unit vector ( � H is unitarily equivalent to ���A��H��
This is very simple to prove� the operator U � H� � H implementing the equivalence is initiallyde�ned on the dense subspace ���A�(� by U���A�(� � ��A�(� this operator is well�de�ned� for���A�(� � � implies ��A�( � � by the GNS�construction It follows from ���� � that U isunitary as a map from H� to UH�� but since ( is cyclic for � the image of U is H Hence U isunitary It is trivial to verify that U intertwines �� and � �
Corollary ��� If the Hilbert spaces H� H� of two cyclic representations ��� �� each contain acyclic vector (� � H� (� � H� and
���A� �� �(�� ���A�(�� � �(�� ���A�(�� �� ���A�
for all A � A then ���A� and ���A� are equivalent�
By ��� the representation �� is equivalent to the GNS�representation ��� � and �� is equivalentto ��� On the other hand� ��� and ��� are induced by the same state� so they must coincide �
���� The Gel�fandNeumark theorem
One of the main results in the theory of C��algebras is
Theorem ���� A C��algebra is isomorphic to a subalgebra of B�H� for some Hilbert space H�
The GNS�construction leads to a simple proof this theorem� which uses the following notion
Denition ���� The universal representation �u of a C��algebra A is the direct sum of allits GNS�representations �� � � S�A�� hence it is de�ned on the Hilbert space Hu � ���S�AH��
�� � ELEMENTARY THEORY OF C��ALGEBRAS
Theorem ���� then follows by taking H � Hu� the desired isomorphism is �u To prove that�u is injective� suppose that �u�A� � � for some A � A By de�nition of a direct sum� this implies���A� � � for all states � Hence ���A�(� � �� hence k ���A�(� k�� �� by ������ this means��A�A� � � for all states �� which implies k A�A k� � by Lemma ���� so that k A k� � by��� �� and �nally A � � by the de�nition of a norm
Being injective� the morphism �u is isometric by Lemma �� �
While the universal representation leads to a nice proof of ����� the Hilbert space Hu isabsurdly large� in practical examples a better way of obtaining a faithful representation alwaysexists For example� the best faithful representation of B�H� is simply its de�ning one
Another consequence of the GNS�construction� or rather of ����� is
Corollary ���� An operator A � A is positive that is A � A�R i� ��A� � � for all cyclic
representations ��
���� Complete positivity
We have seen that a positive map Q �cf De�nition ��� generalizes the notion of a state� in thatthe C in � � A� C is replaced by a general C��algebra B in Q � A � B We would like to see ifone can generalize the GNS�construction It turns out that for this purpose one needs to imposea further condition on Q
We �rst introduce the C��algebra Mn�A� for a given C��algebra A and n � N The elementsof Mn�A� are n � n matrices with entries in A� multiplication is done in the usual way� ie��MN�ij ��
PkMikNkj � with the di�erence that one now multiplies elements of A rather than
complex numbers In particular� the order has to be taken into account The involution in Mn�A�is� of course� given by �M��ij � M�
ji� in which the involution in A replaces the usual complexconjugation in C One may identify Mn�A� with A�Mn�C � in the obvious way
When � is a faithful representation of A �which exists by Theorem ������ one obtains a faithfulrealization �n of Mn�A� on H� C n � de�ned by linear extension of �n�M�vi �� ��Mij�vj � we herelook at elements of H � C n as n�tuples �v�� � � � � vn�� where each vi � H The norm k M k ofM � Mn�A� is then simply de�ned to be the norm of �n�M� Since �n�Mn�A�� is a closed ��algebra in B�H � C n � �because n � ��� it is obvious that Mn�A� is a C��algebra in this normThe norm is unique by Corollary ���� so that this procedure does not depend on the choice of �
Denition ���� A linear map Q � A� B between C��algebras is called completely positiveif for all n � N the map Qn � Mn�A� �Mn�B� de�ned by �Qn�M��ij �� Q�Mij� is positive�
For example� a morphism � is a completely positive map� since when A � B �B in Mn�A�� then��A � � ��B ����B �� which is positive in Mn�B� In particular� any representation of A on H is acompletely positive map from A to B�H�
If we also assume that A and B are unital� and that Q is normalized� we get an interestinggeneralization of the GNS�construction� which is of central importance for quantization theoryThis generalization will appear as the proof of the following Stinespring theorem
Theorem ���� Let Q � A � B be a completely positive map between C��algebras with unit such that Q�I� � I� By Theorem ������ we may assume that B is faithfully represented as asubalgebra B � ��B� B�H� for some Hilbert space H�
There exists a Hilbert space H a representation � of A on H and a partial isometryW � H � H with W �W � I such that
��Q�A�� � W ���A�W �A � A� ������
Equivalently with p �� WW � the target projection of W on H !H �� pH � H and U �
H � !H de�ned as W seen as map not from H to H but as a map from H to !H so that Uis unitary one has
U��Q�A��U�� � p��A�p� ������
���� Complete positivity ��
The proof consists of a modi�cation of the GNS�construction It uses the notion of a partialisometry This is a linear map W � H� � H� between two Hilbert spaces� with the property thatH� contains a closed subspace K� such that �W��W��� � ������ for all ��� � K�� and W � �on K�� Hence W is unitary from K� to WK� It follows that WW � � �K� and W �W � �K� areprojections onto the image and the kernel of W � respectively
We denote elements of H by v� w� with inner product �v� w�
Construction ���� �� De�ne the sesquilinear form � � �� on A�H algebraic tensor prod�uct by sesqui�linear extension of
�A� v�B � w�� �� �v� ��Q�A�B��w�� ������
Since Q is completely positive this form is positive semi�de�nite� denote its null space byN�
�� The form � � �� projects to an inner product � � � on A�H�N� If V � A�H � A�H�N
is the canonical projection then by de�nition
�V�A� v�� V�B � w�� �� �A� v�B � w�� � ������
The Hilbert space H is the closure of A�H�N in this inner product�
�� The representation ��A� is initially de�ned on A�H�N by linear extension of
��A�V�B � w� �� V�AB � w�� ������
this is well�de�ned because ��A�N N� One has the bound
k ��A� k�k A k� ���� �
so that ��A� may be de�ned on all of H by continuous extension of ������ This extensionsatis�es ��A�� � ��A���
�� The map W � H � H de�ned by
Wv �� VI� v ������
is a partial isometry� Its adjoint W � � H � H is given by continuous extension of
W �VA� v � ��Q�A��v� ������
from which the properties W �W � I and ����� follow�
To show that the form de�ned by ������ is positive� we writeXi�j
�Ai � vi� Aj � vj�� �Xi�j
�vi� ��Q�A�iAj��vj�� ������
Now consider the element A of Mn�A� with matrix elements A ij � A�iAj Looking in a faithfulrepresentation �n as explained above� one sees that
�z� A z� �Xi�j
�zi� ��A�iAj�zj� �Xi�j
���Ai�zi� ��Aj�zj� �k Az k�� �
where Az �P
iAizi Hence A � � Since Q is completely positive� it must be that B � de�ned byits matrix elements B ij �� Q�A�iAj�� is positive in Mn�B� Repeating the above argument withA and � replaced by B and �� respectively� one concludes that the right�hand side of ������ ispositive
To prove ���� � one uses ����� in Mn�A� Namely� for arbitrary A�B�� � � � � Bn � A weconjugate the inequality � � A�AIn �k A k� In with the matrix B � whose �rst row is �B�� � � � � Bn��
�� � ELEMENTARY THEORY OF C��ALGEBRAS
and which has zeros everywhere else� the adjoint B � is then the matrix whose �rst column is�B�� � � � � � B
�n�T � and all other entries zero This leads to � � B �A�AB �k A k� B �B Since Q is
completely positive� one has Qn�B �A�AB � �k A k� Qn�B �B � Hence in any representation ��B�and any vector �v�� � � � vn� � H � C n one hasX
i�j
�vi� ��Q�B�i A�ABj��vj� �k A k�
Xi�j
�vi� ��Q�B�i Bj��vj�� ������
With � �P
i VBi � vi� from ������� ������� and ������ one then has
k ��A�� k��Xi�j
�ABi � vi� ABj � vj�� �Xi�j
�vi� ��Q�B�i A�ABj��vj�
�k A k�Xi�j
�vi� ��Q�B�i Bj��vj� �k A k�Xi�j
�Bi � vi� Bj � vj��
�k A k� �VXi
Bi � vi� VXj
Bj � vj� �k A k� k � k� �
To show that W is a partial isometry� use the de�nition to compute
�Wv�Ww� � �VI� v� VI� w� � �I� v� I� w�� � �v� w��
where we used ������ and Q�I� � ITo check ������� one merely uses the de�nition of the adjoint� viz �w�W ��� � �Ww��� for
all w � H and � � H This trivially veri�edTo verify ������� we use ������ and ������ to compute
W ���A�Wv � W ���A�V�I� v� � W �V�A� v� � ��Q�A��v�
Being a partial isometry� one has p � WW � for the projection p onto the image of W � and�in this case� W �W � I for the projection onto the subspace of H on which W is isometric� thissubspace is H itself Hence ������ follows from ������� since
U��Q�A��U�� � W��Q�A��W � � WW ���A�WW � � p��A�p� �
When Q fails to preserve the unit� the above construction still applies� but W is no longer apartial isometry� one rather has kW k��k Q�I� k Thus it is no longer possible to regard H as asubspace of H
If A and perhaps B are non�unital the theorem holds if Q can be extended �as a positivemap� to the unitization of A� such that the extension preserves the unit I �perhaps relative to theunitization of B� When the extension exists but does not preserve the unit� one is in the situationof the previous paragraph
The relevance of Stinespring�s theorem for quantum mechanics stems from the following result
Proposition ���� Let A be a commutative unital C��algebra� Then any positive map Q � A�is completely positive�
By Theorem ��� we may assume that A � C�X� for some locally compact Hausdor� spaceX We may then identify Mn�C�X�� with C�X�Mn�C �� The proof then proceeds in the followingsteps�
� Elements of the form F � where F �x� �P
i fi�x�Mi for fi � C�X� and Mi �Mn�C �� and thesum is �nite� are dense in C�X�Mn�C ��
� Such F is positive i� all fi and Mi are positive
� Positive elements G of C�X�Mn�C �� can be norm�approximated by positive F �s� ie� whenG � � there is a sequence Fk � � such that limk Fk � G
���� Pure states and irreducible representations ��
� Qn�F � is positive when F is positive
� Qn is continuous
If Fk � G � � in C�X�Mn�C �� then Q�G� � limkQ�Fk� is a norm�limit of positive elements�hence is positive
We now prove each of these claims
� Take G � C�X�Mn�C �� and pick � � Since G is continuous� the set
Ox �� fy � X� k G�x� �G�y� k� �g
is open for each x � X This gives an open cover of X � which by the compactness of Xhas a �nite subcover fO
x� � � � �Oxlg A partition of unity subordinate to the given cover
is a collection of continuous positive functions �i � C�X�� where i � �� � � � � l� such that the
support of �i lies in Oxi and
Pli�� �i�x� � � for all x � X Such a partition of unity exists
Now de�ne Fl � C�X�Mn�C �� by
Fl�x� ��
lXi��
�i�x�G�xi�� ������
Since k G�xi��G�x� k� � for all x � Oxi � one has
k Fl�x��G�x� k�klX
i��
�i�x��G�xi��G�x�� k�lX
i��
�i�x� k G�xi��G�x� k�lX
i��
�i�x�� � ��
Here the norm is the matrix norm in Mn�C � Hence
k Fl �G k� supx�X
k Fl�x� �G�x� k� ��
� An element F � C�X�Mn�C �� is positive i� F �x� is positive in Mn�C � for each x � X Inparticular� when F �x� � f�x�M for some f � C�X� and M � Mn�C � then F is positive i�f is positive in C�X� and M is positive in Mn�C � By � �� we infer that F de�ned byF �x� �
Pi fi�x�Mi is positive when all fi and Mi are positive
� When G in item � is positive then each G�xi� is positive� as we have just seen
� On F as speci�ed in ����� one has Qn�F � �P
iQ�fi� �Mi Now each operator Bi �Mis positive in Mn�B� when Bi and M are positive �as can be checked in a faithful repre�sentation� Since Q is positive� it follows that Qn maps each positive element of the formF �
Pi fiMi into a positive member of Mn�B�
� We know from ��� that Q is continuous� the continuity of Qn follows because n ��
A norm�limit A � limnAn of positive elements in a C��algebra is positive� because by �����we have An � B�nBn� and limBn � B exist because of ��� � Finally� A � B�B by continuityof multiplication� ie� by ����� �
���� Pure states and irreducible representations
We return to the discussion at the end of �� One sees that the compact convex sets in theexamples have a natural boundary The intrinsic de�nition of this boundary is as follows
�� � ELEMENTARY THEORY OF C��ALGEBRAS
Denition ���� An extreme point in a convex set K in some vector space is a member �of K which can only be decomposed as
� � ��� " ��� ����� ������
� � ��� �� if �� � �� � �� The collection �eK of extreme points in K is called the extremeboundary of K� An extreme point in the state space K � S�A� of a C��algebra A is called apure state� A state that is not pure is called a mixed state�
When K � S�A� is a state space of a C��algebra we write P�A� or simply P for �eK referredto as the pure state space of A�
Hence the pure states on A � C � C are the points � and � in ��� � � where � is identi�ed withthe functional mapping � '"� to �� whereas � maps it to � The pure states on A � M��C � are thematrices � in ������ for which x� " y� " z� � �� these are the projections onto one�dimensionalsubspaces of C �
More generally� we will prove in ����� that the state space of Mn�C � consists of all positivematrices � with unit trace� the pure state space of Mn�C � then consists of all one�dimensionalprojections This precisely reproduces the notion of a pure state in quantum mechanics The�rst part of De�nition ���� is due to Minkowski� it was von Neumann who recognized that thisde�nition is applicable to quantum mechanics
We may now ask what happens to the GNS�construction when the state � one constructs therepresentation �� from is pure In preparation�
Denition ���� A representation � of a C��algebra A on a Hilbert space H is called irre�ducible if a closed subspace of H which is stable under ��A� is either H or ��
This de�nition should be familiar from the theory of group representations It is a deep fact ofC��algebras that the quali�er �closed� may be omitted from this de�nition� but we will not provethis Clearly� the de�ning representation �d of the matrix algebra MN on CN is irreducible Inthe in�nite�dimensional case� the de�ning representations �d of B�H� on H is irreducible as well
Proposition ���� Each of the following conditions is equivalent to the irreducibility of ��A� onH�
�� The commutant of ��A� in B�H� is f�I j� � C g� in other words ��A��� � B�H� Schur slemma�
�� Every vector ( in H is cyclic for ��A� recall that this means that ��A�( is dense in H�
The commutant ��A�� is a ��algebra in B�H�� so when it is nontrivial it must contain a self�adjoint element A which is not a multiple of I Using Theorem ���� below and the spectraltheorem� it can be shown that the projections in the spectral resolution of A lie in ��A�� if A doesHence when ��A�� is nontrivial it contains a nontrivial projection p But then pH is stable under��A�� contradicting irreducibility Hence %� irreducible � ��A�� � C I&
Conversely� when ��A�� � C I and � is reducible one �nds a contradiction because the projectiononto the alleged nontrivial stable subspace of H commutes with ��A� Hence %��A�� � C I � �irreducible&
When there exists a vector � � H for which ��A�� is not dense in H� we can form the projectiononto the closure of ��A�� By Lemma ���� with M � ��A�� this projection lies in ��A��� so that� cannot be irreducible by Schur�s lemma Hence %� irreducible � every vector cyclic& Theconverse is trivial �
We are now in a position to answer the question posed before ����
Theorem ���� The GNS�representation ���A� of a state � � S�A� is irreducible i� � is pure�
When � is pure yet ���A� reducible� there is a nontrivial projection p � ���A�� by Schur�slemma Let (� be the cyclic vector for �� If p(� � � then Ap(� � pA(� � � for all A � A�
���� Pure states and irreducible representations ��
so that p � � as �� is cyclic Similarly� p�(� � � is impossible We may then decompose� � �� " ��� ����� where � and �� are states de�ned as in ������� with � �� p(�� k p(� k��� �� p�(�� k p�(� k� and � �k p�(� k� Hence � cannot be pure This proves %pure �irreducible&
In the opposite direction� suppose �� is irreducible� with ������ for ��� �� � S�A� and � � ��� � Then ����� � �������� which is positive� hence ����A�A� � ��A�A� for all A � A By ������this yields
j����A�B�j� � �����A�A����B�B� � ��A�A���B�B� ������
for all A�B This allows us to de�ne a quadratic form �ie� a sesquilinear map� �Q on ���A�(� by
�Q����A�(� � ���B�(�� �� ����A�B�� ������
This is well de�ned� when ���A��(� � ���A��(� then ���A� � A����A� � A��� � � by �������so that
j �Q����A��(�� ���B�(��� �Q����A��(�� ���B�(��j� � j�����A� �A���B�j� � �
by ������� in other words� �Q����A��(� � ���B�(�� � �Q����A��(�� ���B�(�� Similarly for BFurthermore� ������ and ������ imply that �Q is bounded in that
j �Q�����j � C k � k k � k� ������
for all ��� � ���A�(�� with C � � It follows that �Q can be extended to all of H� by continuityMoreover� one has
�Q����� � �Q����� ���� �
by ������ with A� A�B and � � ��
Lemma ���� Let a quadratic form �Q on a Hilbert space H be bounded in that ����� holdsfor all ��� � H and some constant C � �� There is a bounded operator Q on H such that�Q����� � ��� Q�� for all ��� � H and k Q k� C� When ����� is satis�ed Q is self�adjoint�
Hold � �xed The map � � �Q����� is then bounded by ������� so that by the Riesz�Fischertheorem there exists a unique vector ( such that �Q����� � �(��� De�ne Q by Q� � ( Theself�adjointness of Q in case that ���� � holds is obvious
Now use ������ to estimate
k Q� k�� �Q�� Q�� � �Q�Q���� � C k Q k k � k��
taking the supremum over all � in the unit ball yields k Q k�� C k Q k�� whence k Q k� C �
Continuing with the proof of ����� we see that there is a self�adjoint operator Q on H� suchthat
����A�(� � Q���B�(�� � ����A�B�� ������
It is the immediate from ������ that �Q� ���C� � � for all C � A Hence Q � ���A��� since �� isirreducible one must have Q � tI for some t � R� hence ������� ������� and ������ show that ��is proportional to �� and therefore equal to � by normalization� so that � is pure �
From ��� we have the
Corollary ���� If ���A��H� is irreducible then the GNS�representation ����A��H�� de�ned byany vector state � corresponding to a unit vector � � H is unitarily equivalent to ���A��H��
Combining this with ���� yields
Corollary ���� Every irreducible representation of a C��algebra comes from a pure state viathe GNS�construction�
� � ELEMENTARY THEORY OF C��ALGEBRAS
A useful reformulation of the notion of a pure state is as follows
Proposition ���� A state is pure i� � � � � � for a positive functional � implies � � t� forsome t � R� �
We assume that A is unital� if not� use �� and ��� For � � � or � � � the claim is obviousWhen � is pure and � � � � �� with � � � � �� then � � ��I� � �� since � � � is positive� hencek � � � k� ��I�� ��I� � �� ��I� Hence ��I� would imply � � �� whereas ��I� � � implies � � ��contrary to assumption Hence �� � ������ ��I�� and ����I� are states� and
� � �� � �
�� ��I�" ��� ��
�
��I�
with � � �� ��I� Since � is pure� by ���� we have � � ��I��Conversely� if ������ holds then � � ��� � � �cf the proof of ������ so that ��� � t� by
assumption� normalization gives t � �� hence �� � � � ��� and � is pure �
The simplest application of this proposition is
Theorem ���� The pure state space of the commutative C��algebra C��X� equipped with therelative w��topology is homeomorphic to X�
In view of Proposition ��� and Theorems ��� and ���� we merely need to establish abijective correspondence between the pure states and the multiplicative functionals on C��X�The case that X is not compact may be reduced to the compact case by passing from A � C��X�to AI � C� !X�� cf �� and ��� etc This is possible because the unique extension of a purestate on C��X� to a state on C� !X� guaranteed by ��� remains pure Moreover� the extension ofa multiplicative functional de�ned in ����� coincides with the extension �I of a state de�ned in������� and the functional � in �� �� clearly de�nes a pure state
Thus we put A � C�X� Let �x � $�C�X�� �cf the proof of ����� and suppose a functional� satis�es � � � � �x Then ker��x� ker���� and ker��� is an ideal But ker��x� is a maximalideal� so when � � � it must be that ker��x� � ker��� Since two functionals on any vector spaceare proportional when they have the same kernel� it follows from ���� that �x is pure
Conversely� let � be a pure state� and pick a g � C�X� with � � g � �X De�ne a functional�g on C�X� by �g�f� �� ��fg� Since ��f� � �g�f� � ��f�� � g��� and � � � � g � �X � onehas � � �g � � Hence �g � t� for some t � R� by ���� In particular� ker��g� � ker��� Itfollows that when f � ker���� then fg � ker��� for all g � C�X�� since any function is a linearcombination of functions for which � � g � �X Hence ker��� is an ideal� which is maximal becausethe kernel of a functional on any vector space has codimension � Hence � is multiplicative byTheorem ��� �
It could be that no pure states exist in S�A�� think of an open convex cone It would follow thatsuch a C��algebra has no irreducible representations Fortunately� this possibility is excluded bythe Krein�Milman theorem in functional analysis� which we state without proof The convexhull co�V � of a subset V of a vector space is de�ned by
co�V � �� f�v " ��� ��w j v� w � V� � � ��� � g� ������
Theorem ����� A compact convex setK embedded in a locally convex vector space is the closureof the convex hull of its extreme points� In other words K � co��eK��
It follows that arbitrary states on a C��algebra may be approximated by �nite convex sumsof pure states This is a spectacular result� for example� applied to C�X� it shows that arbitraryprobability measures on X may be approximated by �nite convex sums of point �Dirac� measuresIn general� it guarantees that a C��algebra has lots of pure states For example� we may now re�neLemma ��� as follows
Theorem ����� For every A � AR and a � �A� there is a pure state �a on A for which�a�A� � a� There exists a pure state � such that j��A�j �k A k�
���� The C��algebra of compact operators ��
We extend the state in the proof of ��� to C��A� I� by multiplicativity and continuity� that is�we put !�a�An� � an etc It follows from ���� that this extension is pure One easily checks thatthe set of all extensions of !�a to A �which extensions we know to be states� see the proof of ����is a closed convex subset Ka of S�A�� hence it is a compact convex set By the Krein�Milmantheorem ����� it has at least one extreme point �a If �a were not an extreme point in S�A�� itwould be decomposable as in ������ But it is clear that� in that case� �� and �� would coincideon C��A� I�� so that �a cannot be an extreme point of Ka �
We may now replace the use of ��� by ����� in the proof of the Gel�fand�Neumark Theorem����� concluding that the universal representation �u may be replaced by �r �� ���P�A�� Wemay further restrict this direct sum by de�ning two states to be equivalent if the correspondingGNS�representations are equivalent� and taking only one pure state in each equivalence class Letus refer to the ensuing set of pure states as �P�A� We then have
A � �r�A� �� ���P�A����A�� ������
It is obvious that the proof of ���� still goes throughThe simplest application of this re�nement is
Proposition ����� Every �nite�dimensional C��algebra is a direct sum of matrix algebras�
For any morphism �� hence certainly for any representation � � �� one has the isomorphism��A� � A� ker��� Since A� ker��� is �nite�dimensional� it must be that ��A� is isomorphic to analgebra acting on a �nite�dimensional vector space Furthermore� it follows from Theorem ����below that ��A��� � ��A� in every �nite�dimensional representation of A� upon which �����implies that ��A� must be a matrix algebra �as B�H� is the algebra of n�n matrices for H � C n �Then apply the isomorphism ���� �
���� The C�algebra of compact operators
It would appear that the appropriate generalization of the C��algebra Mn�C � of n � n matricesto in�nite�dimensional Hilbert spaces H is the C��algebra B�H� of all bounded operators on HThis is not the case For one thing� unlike Mn�C � �which� as will follow from this section� hasonly one irreducible representation up to equivalence�� B�H� has a huge number of inequivalentrepresentations� even when H is separable� most of these are realized on non�separable Hilbertspaces
For example� it follows from ��� that any vector state � on B�H� de�nes an irreduciblerepresentation of B�H� which is equivalent to the de�ning representation On the other hand� weknow from ����� and the existence of bounded self�adjoint operators with continuous spectrum�such as any multiplication operator on L��X�� where X is connected�� that there are many otherpure states whose GNS�representation is not equivalent to the de�ning representation � Namely�when A � B�H� and a � �A�� but a is not in the discrete spectrum of A as an operator on H�ie� there is no eigenvector �a � H for which A�a � a�a�� then ��a cannot be equivalent to �For it is easy to show from ���� � that (�a � H�a is an eigenvector of ��a�A� with eigenvaluea In other words� a is in the continuous spectrum of A � ��A� but in the discrete spectrum of��a�A�� which excludes the possibility that ��A� and ��a�A� are equivalent �as the spectrum isinvariant under unitary transformations�
Another argument against B�H� is that it is non�separable in the nom�topology even when His separable The appropriate generalization of Mn�C � to an in�nite�dimensional Hilbert space Hturns out to be the C��algebra B��H� of compact operator on H In non�commutative geometryelements of this C��algebra play the role of in�nitesimals� in general� B��H� is a basic buildingblock in the theory of C��algebras This section is devoted to an exhaustive study of this C��algebra
Denition ���� Let H be a Hilbert space� The ��algebra Bf �H� of nite�rank operators onH is the �nite linear span of all �nite�dimensional projections on H� In other words an operatorA � B�H� lies in Bf �H� when AH �� fA�j� � Hg is �nite�dimensional�
�� � ELEMENTARY THEORY OF C��ALGEBRAS
The C��algebra B��H� of compact operators on H is the norm�closure of Bf �H� in B�H��in other words it is the smallest C��algebra of B�H� containing Bf �H�� In particular the normin B��H� is the operator norm ���� An operator A � B�H� lies in B��H� when it can beapproximated in norm by �nite�rank operators�
It is clear that Bf �H� is a ��algebra� since p� � p for any projection p The third item in the nextproposition explains the use of the word �compact� in the present context
Proposition ���� �� The unit operator I lies in B��H� i� H is �nite�dimensional�
�� The C��algebra B��H� is an ideal in B�H��
�� If A � B��H� then AB� is compact in H with the norm�topology� Here B� is the unit ballin H i�e� the set of all � � H with k � k� ��
Firstly� for any sequence �or net� An � Bf �H� we may choose a unit vector �n � �AnH��Then �An � I�� � ��� so that k �An � I�� k� � Hence supk�k�� k �An � I�� k� �� hencek An � Ik� � is impossible by de�nition of the norm ���� in B�H� �hence in B��H��
Secondly� when A � Bf �H� and B � B�H� then AB � Bf �H�� since ABH � AH But sinceBA � �A�B���� and Bf �H� is a ��algebra� one has A�B� � Bf �H� and hence BA � Bf �H�Hence Bf �H� is an ideal in B�H�� save for the fact that it is not norm�closed �unless H has �nitedimension� Now if An � A then AnB � AB and BAn � BA by continuity of multiplication inB�H� Hence B��H� is an ideal by virtue of its de�nition
Thirdly� note that the weak topology onH �in which �n � � i� ����n� � ����� for all � � H�is actually the w��topology under the duality of H with itself given by the Riesz�Fischer theoremHence the unit ball B� is compact in the weak topology by the Banach�Alaoglu theorem So if wecan show that A � B��H� maps weakly convergent sequences to norm�convergent sequences� thenA is continuous from H with the weak topology to H with the norm�topology� since compactnessis preserved under continuous maps� it follows that AB� is compact
Indeed� let �n � � in the weak topology� with k �n k� � for all n Since
k � k�� ����� � limn
����n� �k � k k �n k� ��
one has k � k� � Given � �� choose Af � Bf �H� such that k A � Af k� ���� and putp �� �AfH � the �nite�dimensional projection onto the image of Af Then
k A�n �A� k�k �A�Af ��n " �A�Af �� " Af ��n ��� k� �
�� " �
��" k Af k k p��n ��� k �
Since the weak and the norm topology on a �nite�dimensional Hilbert space coincide� they coincideon pH� so that we can �nd N such that k p��n ��� k� ��� for all n N Hence k A�n �A� k� � �
Corollary ���� A self�adjoint operator A � B��H� has an eigenvector �a with eigenvalue asuch that jaj �k A k�
De�ne fA � B� � R by fA��� ��k A� k� When �n � � weakly with k �n k� �� then
jfA��n��fA���j � j��n� A�A��n���������n� A
�A��j �k A�A��n��� k "j����n� A�A��j�
The �rst term goes to zero by the proof of ����� �noting that A�A � B��H��� and the second goesto zero by de�nition of weak convergence Hence fA is continuous Since B� is weakly compact�fA assumes its maximum at some �a This maximum is k A k� by ���� Now the Cauchy�Schwarz inequality with � � � gives k A� k�� ��� A�A�� �k A�A� k� with equality i� A�A� isproportional to � Hence when A� � A the property k A k��k A�a k� with k �a k� � impliesA��a � a��a� where a� �k A k� The spectral theorem or the continuous functional calculus with
f�A�� �pA� � A implies A�a � a�a Clearly jaj �k A k �
���� The C��algebra of compact operators ��
Theorem ���� A self�adjoint operator A � B�H� is compact i� A �P
i ai��i norm�convergentsum where each eigenvalue ai has �nite multiplicity� Ordering the eigenvalues so that ai � ajwhen i j one has limi�� jaij � �� In other words the set of eigenvalues is discrete and canonly have � as a possible accumulation point�
This ordering is possible because by ���� there is a largest eigenvalueLet A � B��H� be self�adjoint� and let p be the projection onto the closure of the linear
span of all eigenvectors of A As in Lemma ��� one sees that �A� p � �� so that �pA�� � pAHence p�A � �I� p�A is self�adjoint� and compact by ����� By ���� the compact self�adjointoperator p�A has an eigenvector� which must lie in p�H� and must therefore be an eigenvectorof A in p�H By assumption this eigenvector can only be zero Hence k p�A k� � ����� whichimplies that A restricted to p�H is zero� which implies that all vectors in p�H are eigenvectorswith eigenvalue zero This contradicts the de�nition of p�H unless p�H � � This proves %Acompact and self�adjoint � A diagonalizable&
Let A be compact and self�adjoint� hence diagonalizable Normalize the eigenvectors �i �� �ai
to unit length Then limi������i� � � for all � � H� since the �i form a basis� so that
����� �Xi
j����i�j�� ������
which clearly converges Hence �i � � weakly� so k A�i k� jaij � � by �the proof of������ Hence limi�� jaij � � This proves %A compact and self�adjoint � A diagonalizablewith limi�� jaij � �&
Let now A be self�adjoint and diagonalizable� with limi�� jaij � � For N � � and � � Hone then has
k �A�NXi��
ai��i �� k��k�X
i�N��
ai��i����i k���X
i�N��
jaij� j����i�j� � jaN j��X
i�N��
j����i�j��
Using ������� this is� jaN j������� so that limN�� k A�PNi�� ai��i k� �� because limN�� jaN j �
� Since the operatorPN
i�� ai��i is clearly of �nite rank� this proves that A is compact Hence%A self�adjoint and diagonalizable with limi�� jaij � � � A compact&
Finally� when A is compact its restriction to any closed subspace of H is compact� which by����� proves the claim about the multiplicity of the eigenvalues �
We now wish to compute the state space of B��H� This involves the study of a number ofsubspaces of B�H� which are not C��algebras� but which are ideals of B�H�� except for the factthat they are not closed
Denition ���� The Hilbert�Schmidt norm k A k� of A � B�H� is de�ned by
k A k����Xi
k Aei k�� ������
where feigi is an arbitrary basis of H� the right�hand side is independent of the choice of the basis�The Hilbert�Schmidt class B��H� consists of all A � B�H� for which k A k����
The trace norm k A k� of A � B�H� is de�ned by
k A k���k �A�A��
� k��� ������
where �A�A��
� is de�ned by the continuous functional calculus� The trace class B��H� consistsof all A � B�H� for which k A k����
To show that ������ is independent of the basis� we take a second basis fuigi� with corre�sponding resolution of the identity I�
Pi�ui �weakly� Aince I�
Pi�ei we then have
k A k����Xi�j
�ej �ui��ui� A�Aej� �
Xi�j
�A�Aui� ej��ej �ui� �Xi
k Aui k� �
�� � ELEMENTARY THEORY OF C��ALGEBRAS
If A � B��H� then
TrA ��Xi
�ei� Aei� ������
is �nite and independent of the basis �when A �� B��H�� it may happen that TrA depends onthe basis� it may even be �nite in one basis and in�nite in another� Conversely� it can be shownthat A � B��H� when Tr�A ��� where Tr� is de�ned in terms of the decomposition ������ byTr�A �� TrA�� � TrA�� " iTrA��� � iTrA��� For A � B��H� one has Tr�A � TrA One alwayshas the equalities
k A k� � Tr jAj� ������
k A k� � Tr jAj� � TrA�A� ������
wherejAj ��
pA�A� ���� �
In particular� when A � � one simply has k A k�� TrA� which does not depend on the basis�whether or not A � B��H� The properties
TrA�A � TrAA� ������
for all A � B�H�� andTrUAU� � TrA ������
for all positive A � B�H� and all unitaries U � follow from ������ by manipulations similar tothose establishing the basis�independence of ������ Also� the linearity property
Tr �A " B� � TrA " TrB ������
for all A�B � B��H� is immediate from ������It is easy to see that the Hilbert�Schmidt norm is indeed a norm� and that B��H� is complete
in this norm The corresponding properties for the trace norm are nontrivial �but true�� and willnot be needed In any case� for all A � B�H� one has
k A k � k A k�� ������
k A k � k A k� � ������
To prove this� we use our old trick� although k B k�k B� k for all unit vectors �� for every � �
there is a � � H of norm � such that k B k��k B� k� "� Put B � �A�A��
� � and note that
k �A�A��
� k��k A k by ��� � Completing � to a basis feigi� we have
k A k�k �A�A��
� k��k �A�A��
� � k� "� �Xi
k �A�A��
� ei k� "� �k A k� "��
Letting �� � then proves ������ The same trick with k A k�k A� k "� establishes ������The following decomposition will often be used
Lemma ���� Every operator A � B�H� has a polar decomposition
A � U jAj� ������
where jAj �pA�A cf� ����� and U is a partial isometry with the same kernel as A�
First de�ne U on the range of jAj by U jAj� �� A� Then compute
�U jAj�� U jAj�� � �A�� A�� � ��� A�A�� � ��� jAj��� � �jAj�� jAj���
Hence U is an isometry on ran�jAj� In particular� U is well de�ned� for this property implies thatif jAj�� � jAj�� then U jAj�� � U jAj�� Then extend U to the closure of ran�jAj� by continuity�and put U � � on ran�jAj�� One easily veri�es that
jAj � U�A� ������
and that U�U is the projection onto the closure of ran�jAj�� whereas UU� is the projection ontothe closure of ran�A� �
���� The C��algebra of compact operators ��
Proposition ���� One has the inclusions
Bf �H� B��H� B��H� B��H� B�H�� ������
with equalities i� H is �nite�dimensional�
We �rst show that B��H� B��H� Let A � B��H� SinceP
i�ei� jAjei� ��� for every � �we can �nd N��� such that
Pi�N��ei� jAjei� � � Let pN� be the projection onto the linear
span of all ei� i N��� Using ��� � and ������� we have
k jAj �� pN� k��k pN�jAjpN� k�k pN�jAjpN� k�� ��
so that jAj �� p�N� � jAj �� in the operator�norm topology Since the star is norm�continuous by
������ this implies p�N�jAj�
� � jAj �� Now p�N�jAj�
� obviously has �nite rank for every � �� so
that jAj �� is compact by De�nition ���� Since A � U jAj �� jAj �� by ������� Proposition �����implies that A � B��H�
The proof that B��H� B��H� is similar� this time we have
k jAjpN� k��k pN�jAj�pN� k�k pN�jAj�pN� k�� ��
so that jAjp�N� � jAj� with the same conclusion
Finally� we use Theorem ���� to rewrite ������ and ������ as
k A k� �Xi
ai�
k A k� �Xi
a�i � ������
where the ai are the eigenvalues of jAj This immediately gives
k A k��k A k�� ���� �
implying B��H� B��H�Finally� the claim about proper inclusions is trivially established by producing examples on the
basis of ���� and ������ �
The chain of inclusions ������ is sometimes seen as the non�commutative analogue of
�c�X� ���X� ���X� ���X� ���X��
where X is an in�nite discrete set Since ���X� � ���X�� and ���X� � ���X�� � ���X���� thisanalogy is strengthened by the following result
Theorem ���� One has B��H�� � B��H� and B��H�� � B��H��� � B�H� under the pairing
���A� � Tr �A � �A���� ������
Here �� � B��H�� is identi�ed with � � B��H� and �A � B��H�� is identi�ed with A � B�H��
The basic ingredient in the proof is the following lemma� whose proof is based on the fact thatB��H� is a Hilbert space in the inner product
�A�B� �� TrA�B� ������
To show that this is well de�ned� use ���� and ������
Lemma ���� For � � B��H� and A � B�H� one has
jTrA�j �k A k k � k� � ������
�� � ELEMENTARY THEORY OF C��ALGEBRAS
Using ������ for � and ���� for the inner product ������� as well as ������ and ������� weestimate
jTrA�j� � jTrAU j�j �� j�j �� j � j��AU j�j �� ��� j�j �� �j
� k j�j �� k�� k �AU j�j �� �� k���k � k� Tr �j�j ��U�A�AU j�j �� ��
Now observe that if � � A� � A� then TrA� � TrA� for all A�� A� � B��H�� since on accountof ���� one has A� � A� i� all eigenvalues of A� are � all eigenvalues of A� Then use ������
From ����� we have j�j ��U�A�AU j�j �� �k AU k� �� so from the above insight we arrive at
Tr �j�j ��U�A�AU j�j �� � �k � k� k AU k��k A k��
since U is a partial isometry Hence we have ������ �
We now prove B��H�� � B��H� It is clear from ���� that
B��H� B��H��� ��� ��
with
k �� k�k � k� � ��� ��
To prove that B��H�� B��H�� we use ������ For �� � B��H�� and A � B��H� B��H� wetherefore have
j���A�j � k �� k k A k�k �� k k A k� �Hence �� � B��H��� sinceB��H� is a Hilbert space� by Riesz�Fischer there is an operator � � B��H�such that ���A� � Tr �A for all A � B��H� In view of ������� we need to sharpen � � B��H�to � � B��H� To do so� choose a �nite�dimensional projection p� and note that pj�j � Bf �H� B��H�� the presence of p even causes the sum in ������ to be �nite in a suitable basis Now usethe polar decomposition � � U j�j with ������ to write
Tr pj�j � Tr pU�� � Tr �pU� � ���pU���
changing the order inside the trace is justi�ed by naive arguments� since the sum in ������ is�nite Using the original assumption �� � B��H��� we have
jTr pj�j j � k �� k k pU� k�k �� k k p k�k �� k ��� ��
since U is a partial isometry� whereas k p k� � in view of ��� � and p � p� � p� Now choosea basis of H� and take p to be the projection onto the subspace spanned by the �rst N elements�from ������ and ��� �� we then have
jTr pj�j j � jNXi��
�ei� j�jei�j � k �� k �
It follows that the sequence sN �� jPNi���ei� j�jei�j is bounded� and since it is positive it must
have a limit By ��� �� and ������ this means that k � k��k �� k� so that � � B��H�� henceB��H�� B��H� Combining this with ��� �� and ��� ��� we conclude that B��H�� � B��H�and k � k��k �� k
We turn to the proof of B��H�� � B�H� It is clear from ���� that B�H� B��H��� with
k �A k�k A k � ��� ��
To establish the converse� pick �A � B��H�� and ��� � H� and de�ne a quadratic form QA on Hby
QA����� �� �A�j� � �j�� ��� ��
���� The C��algebra of compact operators ��
Here the operator j� � �j is de�ned by j� � �j( �� ���(�� For example� when � hasunit length� j� � �j is the projection �� � and in general j� � �j �k � k� �� Note that�j� � �j�� � j� � �j� so that
j j� � �j j �p
�j� � �j��j� � �j �p
�����j� � �j �k � k k � k �� �
Since� for any projection p� the number Tr p is the dimension of pH �take a basis whose elementslie either in pH or in p�H�� we have Tr �� � � Hence from ������ we obtain
k j� � �j k��k � k k � k � ��� ��
Since �A � B��H�� by assumption� one has
j �A�j� � �j�j �k �A k k j� � �j k� � ��� �
Combining ��� �� ��� ��� and ��� ��� we have
jQA�����j �k �A k k � k k � k � ��� ��
Hence by Lemma ���� and ��� �� there is an operator A� with
k A k�k �A k� ��� ��
such that �A�j� � �j� � ��� A�� Now note that ��� A�� � Tr j� � �jA� this follows byevaluating ������ over a basis containing k � k�� j� Hence �A�j� � �j� � Tr j� � �jAExtending this equation by linearity to the span Bf �H� of all j� � �j� and subsequently by
continuity to B��H�� we obtain �A� � Tr �A Hence B��H�� B�H�� so that� with ��� ���we obtain B��H�� � B�H� Combining ��� �� and ��� ��� we �nd k A k�k �A k� so that theidenti�cation of B��H�� with B�H� is isometric �
Corollary ����� �� The state space of the C��algebra B��H� of all compact operators onsome Hilbert space H consists of all density matrices where a density matrix is an element� � B��H� which is positive � � � and has unit trace Tr � � ��
�� The pure state space of B��H� consists of all one�dimensional projections�
�� The C��algebra B��H� possesses only one irreducible representation up to unitary equiva�lence namely the de�ning one�
Diagonalize � �P
i pi��i � cf ���� and ���� Using A � ��i � which is positive� the condition���A� � � yields pi � � Conversely� when all pi � � the operator � is positive The normalizationcondition k �� k�k � k��
Ppi � � �see �� � and ������� yields ������
The next item ������ is then obvious from ����Finally� ������ follows from ������ and Corollaries ���� and ��� �
Corollary ������ is one of the most important results in the theory of C��algebras Appliedto the �nite�dimensional case� it shows that the C��algebra Mn�C � of n�n matrices has only oneirreducible representation
The opposite extreme to a pure state on B��H� is a faithful state ��� for which by de�nitionthe left�ideal N� de�ned in ������ is zero In other words� one has TrA�A � for all A � �
Proposition ����� The GNS�representation �� corresponding to a faithful state �� on B��H� isunitarily equivalent to the representation ����B��H�� on the Hilbert space B��H� of Hilbert�Schmidtoperators given by left�multiplication i�e�
����A�B �� AB� ��� ��
�� � ELEMENTARY THEORY OF C��ALGEBRAS
It is obvious from ������ that for A � B�H� and B � B��H� one has
k AB k��k A k k B k�� ������
so that the representation ��� �� is well�de�ned �even for A � B�H� rather than merely A �B��H�� Moreover� when A�B � B��H� one has
TrAB � TrBA� ������
This follows from ������ and the identity
AB � �
�
�Xn��
in�B " inA����B " inA��� ������
When � � B��H� and � � � then ���� � B��H�� see ������ and ������ It is easily seen that ����
is cyclic for ����B��H�� when �� is faithful Using ������ and ������ we compute
������ ����A������ � Tr ��������A����� � Tr �A � ���A��
The equivalence between �� and ��� now follows from �� or ��� �
For an alternative proof� use the GNS construction itself The map A� A����� with � � B��H��maps B��H� into B��H�� and if �� is faithful the closure �in norm derived from the inner product������� of the image of this map is B��H�
���� The double commutant theorem
The so�called double commutant theorem was proved by von Neumann in ����� and remains acentral result in operator algebra theory For example� although it is a statement about vonNeumann algebras� it controls the �ir�reducibility of representations Recall that the commutantM� of a collection M of bounded operators consists of all bounded operators which commute withall elements of M� the bicommutant M�� is �M���
We �rst give the �nite�dimensional version of the theorem� this is already nontrivial� and itsproof contains the main idea of the proof of the in�nite�dimensional case as well
Proposition ���� Let M be a ��algebra and hence a C��algebra in Mn�C � containing I heren ��� Then M�� � M�
The idea of the proof is to take n arbitrary vectors ��� � � � ��n in C n � and� given A � M���construct a matrix A� � M such that A�i � A��i for all i � �� � � � � n Hence A � A� � M Wewill write H for C n
Choose some � � �� � H� and form the linear subspace M� of H Since H is �nite�dimensional� this subspace is closed� and we may consider the projection p � �M� onto thissubspace By Lemma ��� one has p � M� Hence A � M�� commutes with p Since I� M� wetherefore have � � I� � M�� so � � p�� and A� � Ap� � pA� �M� Hence A� � A�� forsome A� �M
Now choose ��� � � � ��n � H� and regard �� '" � � � '"�n as an element of Hn �� �nH � H� C n�the direct sum of n copies of H�� where �i lies in the i�th copy Furthermore� embed M inB�Hn� � Mn�B�H�� by A � ��A� �� AIn �where In is the unit in Mn�B�H���� this is thediagonal matrix in Mn�B�H�� in which all diagonal entries are A
Now use the �rst part of the proof� with the substitutionsH � Hn�M� ��M�� A� A �� ��A��and � � �� '" � � � '"�n Hence given �� '" � � � '"�n and ��A� � ��M� there exists A � � ��M��� suchthat
��A���� '" � � � '"�n� � A � ��� '" � � � '"�n�� ������
For arbitrary B � Mn�B�H��� compute ��B � ��A� �ij � �Bij � A Hence ��M�� � Mn�M�� It iseasy to see that Mn�M��� � Mn�M���� so that
��M��� � ��M���� ������
���� The double commutant theorem ��
Therefore� A � � ��A�� for some A� �M Hence ������ reads A�i � A��i for all i � �� � � � � n �
As it stands� Proposition ���� is not valid when Mn�C � is replaced by B�H�� where dim�H� �� To describe the appropriate re�nement� we de�ne two topologies on B�H� which are weakerthan the norm�topology we have used so far �and whose de�nition we repeat for convenience�
Denition ���� � The norm�topology onB�H� is de�ned by the criterion for convergenceA� � A i� k A� �A k� �� A basis for the norm�topology is given by all sets of the form
On �A� �� fB � B�H�j k B �A k� �g� ������
where A � B�H� and � ��
� The strong topology on B�H� is de�ned by the convergence A� � A i� k �A��A�� k� �for all � � H� A basis for the strong topology is given by all sets of the form
Os�A���� � � � ��n� �� fB � B�H�j k �B �A��i k� � �i � �� � � � � ng� ���� �
where A � B�H� ��� � � � ��n � H and � ��
� The weak topology on B�H� is de�ned by the convergence A� � A i� j��� �A��A���j � �for all � � H� A basis for the weak topology is given by all sets of the form
Ow �A���� � � � ��n���� � � � ��n� �� fB � B�H�j j��i� �A� �A��i�j � � �i � �� � � � � ng�
������where A � B�H� ��� � � � ��n���� � � � ��n � H and � ��
These topologies should all be seen in the light of the general theory of locally convex topologicalvector spaces These are vector spaces whose topology is de�ned by a family fp�g of semi�norms�recall that a semi�norm on a vector space V is a function p � V � R satisfying ����� �� and � Anet fv�g in V converges to v in the topology generated by a given i� p��v� � v� � � for all �
The norm�topology is de�ned by a single semi�norm� namely the operator norm� which is evena norm Its open sets are generated by ��balls in the operator norm� whereas the strong and theweak topologies are generated by �nite intersections of ��balls de�ned by semi�norms of the formps��A� ��k A� k and pw�� �A� �� j��� A��j� respectively The equivalence between the de�nitionsof convergence stated in ���� and the topologies de�ned by the open sets in question is given intheory of locally convex topological vector spaces
The estimate ���� shows that norm�convergence implies strong convergence Using the Cauch�Schwarz inequality ���� one sees that strong convergence implies weak convergence In otherwords� the norm topology is stronger than the strong topology� which in turn is stronger than theweak topology
Theorem ���� Let M be a ��algebra in B�H� containing I� The following are equivalent�
�� M�� � M�
�� M is closed in the weak operator topology�
�� M is closed in the strong operator topology�
It is easily veri�ed from the de�nition of weak convergence that the commutantN� of a ��algebraN is always weakly closed� for if A� � A weakly with all A� � N� and B � N� then
��� �A�B �� � ��� AB��� �B��� A�� � lim�
��� A�B��� �B��� A��� � lim�
��� �A�� B �� � ��
If M�� � M then M � N� for N � M�� so that M is weakly closed Hence %� � �&Since the weak topology is weaker than the strong topology� %� � �& is trivialTo prove %� � �&� we adapt the proof of ���� to the in�nite�dimensional situation Instead
of M�� which may not be closed� we consider its closure M�� so that p � �M� Hence A �M��
� � HILBERT C��MODULES AND INDUCED REPRESENTATIONS
implies A �M�� in other words� for every � � there is an A �M such that k �A�A�� k� �For Hn this means that
k ��A�A���� '" � � � '"�n� k��nXi��
k �A�A��i k�� ���
Noting the inclusion
fnXi��
k �A�B��i k�� ��g Os �A���� � � � ��n�
�cf ���� ��� it follows that A � A for � � � Since all A � M and M is strongly closed� thisimplies that A � M� so that M�� M With the trivial inclusion M M��� this proves thatM�� � M �
� Hilbert C��modules and induced representations
��� Vector bundles
This chapter is concerned with the �non�commutative analogue� of a vector bundle Let us �rstrecall the notion of an ordinary vector bundle� this is a special case of the following
Denition ��� A bundle B�X�F� �� consists of topological spaces B the total space X thebase F the typical ber and a continuous surjection � � P� X with the following property�each x � X has a neighbourhood N� such that there is a homeomorphism �� � ����N�� � N��F �X � F for which � � �X � �� where �X � X � F � X is the projection onto the �rst factor�
The maps �� are called local trivializations We factorize �� � ��� �F� �� so that �F� restrictedto ����x� provides a homeomorphism between the latter and the typical �ber F Each subset����x� is called a ber of B One may think of B as X with a copy of F attached at each point
The simplest example of a bundle over a base X with typical �ber F is the trivial bundleB � X � F � with ��x� f� �� x According to the de�nition� any bundle is locally trivial in thespeci�ed sense
Denition ��� A vector bundle is a bundle in which
�� each �ber is a �nite�dimensional vector space such that the relative topology of each �bercoincides with its topology as a vector space�
�� each local trivialization �F� � ����x� � F where x � N� is linear�
A complex vector bundle is a vector bundle with typical �ber Cm for some m � N�We will generically denote vector bundles by the letter V� with typical �ber F � V The
simplest vector bundle over X with �ber V � C n is the trivial bundle V � X � C n Thisbundle leads to possibly nontrivial sub�bundles� as follows Recall the de�nition of Mn�A� in���� specialized to A � C�X� in the proof of ���� If X is a compact Hausdor� space� thenMn�C�X�� � C�X�Mn�C �� is a C��algebra Let X in addition be connected One should verifythat a matrix�valued function p � C�X�Mn�C �� is an idempotent �that is� p� � p� i� each p�x� isan idempotent in Mn�C � Such an idempotent p de�nes a vector bundle Vp� whose �ber above x is����x� �� p�x�C n The space Vp inherits a topology and a projection � �onto the �rst co�ordinate�from X�C n � relative to which all axioms for a vector bundle are satis�ed Note that the dimensionof p�x� is independent of x� because p is continuous and X is connected
The converse is also true
Proposition ��� Let V be a complex vector bundle over a connected compact Hausdor� spaceX with typical �ber Cm � There is an integer n � m and an idempotent p � C�X�Mn�C �� suchthat V X � C n with ����x� � p�x�C n �
��� Vector bundles ��
The essence of the proof is the construction of a complex vector bundle V� such that V � V� istrivial �where the direct sum is de�ned �berwise�� this is the bundle X � C n
Following the philosophy of non�commutative geometry� we now try to describe vector bundlesin terms of C��algebras The �rst step is the notion of a section of V� this is a map � � X � V
for which ����x�� � x for all x � X In other words� a section maps a point in the base space intothe �ber above the point Thus one de�nes the space ��V� of all continuous sections of V This isa vector space under pointwise addition and scalar multiplication �recall that each �ber of V is avector space� Moreover� when X is a connected compact Hausdor� space� ��V� is a right�modulefor the commutative C��algebra C�X�� one obtains a linear action �R of C�X� on ��V� by
�R�f���x� �� f�x���x�� ����
Since C�X� is commutative� this is� of course� a left�action as wellFor example� in the trivial case one has the obvious isomorphisms
��X � Cm � � C�X� Cm � � C�X�� Cm � �mC�X�� ����
A fancy way of saying this is that ��X � Cm � is a nitely generated free module for C�X�Here a free �right�� module E for an algebra A is a direct sum E � �nA of a number of copies ofA itself� on which A acts by right�multiplication� ie�
�R�B�A� � � � ��An �� A�B � � � ��AnB� ����
If this number is �nite one says that the free module is �nitely generatedWhen V is non�trivial� one obtains ��V� as a certain modi�cation of a �nitely generated free
module for C�X� For any algebraA� and idempotent p �Mn�A�� the action of p on�nA commuteswith the action by A given by right�multiplication on each component Hence the vector spacep �m A is a right� A�module� called projective When m � �� one calls p �m A a nitelygenerated projective module for A
In particular� when V � X � C n and Vp is the vector bundle described prior to ���� we seethat
��Vp� � p�n C�X� ����
under the obvious �right�� action of C�X�This lead to the Serre�Swan theorem
Theorem ��� Let X be a connected compact Hausdor� space� There is a bijective corre�spondence between complex vector bundles V over X and �nitely generated projective modulesE�V� � ��V� for C�X��
This is an immediate consequence of Proposition ���� any vector bundle is of the form Vp�leading to ��Vp� as a �nitely generated projective C�X��module by ���� Conversely� given such amodule p�n C�X�� one has p � C�X�Mn�C ��� and thereby a vector bundle Vp as described priorto ��� �
Thus we have achieved our goal of describing vector bundles over X purely in terms of conceptspertinent to the C��algebra C�X� Let us now add further structure
Denition ��� A Hermitian vector bundle is a complex vector bundle V with an innerproduct � � �x de�ned on each �ber ����x� which continuously depends on x� More precisely forall ��� � ��V� the function x� ���x����x��x lies in C�X��
Using the local triviality of V and the existence of a partition of unity� it is easily shown thatany complex vector bundle over a paracompact space can be equipped with such a Hermitianstructure Describing the bundle as Vp� a Hermitian structure is simply given by restricting thenatural inner product on each �ber C n of X � C n to Vp One may then choose the idempotentp � C�X� C n � so as to be a projection with respect to the usual involution on C�X� C n � �ie� one
�� � HILBERT C��MODULES AND INDUCED REPRESENTATIONS
has p� � p in addition to p� � p� Any other Hermitian structure on Vp may be shown to beequivalent to this canonical one
There is no reason to restrict the dimension of the �bers so as to be �nite�dimensional AHilbert bundle is de�ned by replacing ��nite�dimensional vector space� in ���� by �Hilbertspace�� still requiring that all �bers have the same dimension �which may be in�nite� A Hilbertbundle with �nite�dimensional �bers is evidently the same as a Hermitian vector bundle Thesimplest example of a Hilbert bundle is a Hilbert space� seen as a bundle over the base spaceconsisting of a single point
The following class of Hilbert bundles will play a central role in the theory of induced grouprepresentations
Proposition ��� Let H be a closed subgroup of a locally compact group G and take a uni�tary representation U of H on a Hilbert space H� Then H acts on G � H by h � �x� v� ��xh��� U�h�v� and the quotient
H �� G�H H � �G�H��H ����
by this action is a Hilbert bundle over X � G�H with projection
���x� v H � �� �x H �� �
and typical �ber H�
Here �x� v H is the equivalence class in G �H H of �x� v� � G � H� and �x H � xH is theequivalence class in G�H of x � G Note that the projection � is well de�ned
The proof relies on the fact that G is a bundle over G�H with projection
��x� � �x H ����
and typical �ber H This fact� whose proof we omit� implies that every q � G�H has a neighbour�hood N�� so that �� � ��� �H� � � ����N�� � N� �H is a di�eomorphism� which satis�es
�H� �xh� � �H� �x�h� ����
This leads to a map �� � ��� �N�� � N� �H� given by ����x� v H � �� ��x H � U��H� �x��v� Thismap is well de�ned because of ����� and is a local trivialization of G�HH All required propertiesare easily checked �
��� Hilbert C�modules
What follows generalizes the notion of a Hilbert bundle in such a way that the commutative C��algebra C�X� is replaced by an arbitrary C��algebra B This is an example of the strategy ofnon�commutative geometry
Denition ��� A Hilbert C��module over a C��algebra B consists of
� A complex linear space E�� A right�action �R of B on E i�e� �R maps B linearly into the space of all linear operatorson E and satis�es �R�AB� � �R�B��R�A� for which we shall write �B �� �R�B�� where� � E and B � B�
� A sesquilinear map h � iB � E �E � B linear in the second and anti�linear in the �rst entry satisfying
h���i�B � h���iB� ����
h���BiB � h���iBB� �����
h���iB � �� �����
h���iB � � � � � �� �����
for all ��� � E and B � B�
��� Hilbert C��modules ��
The space E is complete in the norm
k � k��k h���iB k �
� � �����
We say that E is a Hilbert B�module and write E � B�
One checks that ����� is indeed a norm� k � k� equals supf��h���iB�g� where the supremumis taken over all states � on B Since each map � � p��h���iB� is a semi�norm �ie� a normexcept for positive de�niteness� by ������ the supremum is a semi�norm� which is actually positivede�nite because of Lemma ��� and �����
The B�action on E is automatically non�degenerate� the property �B � � for all B � B impliesthat h���iBB � � for all B� hence h���iB � � �when B is unital this is follows by taking B � I�otherwise one uses an approximate unit in B�� so that � � � by �����
When all conditions in ��� are met except ������ so that k � k de�ned by ����� is only asemi�norm� one simply takes the quotient of E by its subspace of all null vectors and completes�obtaining a Hilbert C��module in that way
It is useful to note that ���� and ����� imply that
h�B��iB � B�h���iB� �����
Example ��� �� Any C��algebra A is a A�module A� A over itself with hA�BiA �� A�B�Note that the norm ���� coincides with the C��norm by �����
�� Any Hilbert space H is a Hilbert C �module H� C in its inner product�
�� Let H be a Hilbert bundle H over a compact Hausdor� space X� The space of continuoussections E � ��H� of H is a Hilbert C��module ��H� � C�X� over B � C�X�� for ��� ����H� the function h���iC�X is de�ned by
h���iC�X � x� ���x����x��x� �����
where the inner product is the one in the �ber ����x�� The right�action of C�X� on ��H� isde�ned by ����
In the third example the norm in ��H� is k � k� supx k ��x� k� where k ��x� k� ���x����x���
�x �
so that it is easily seen that E is completeMany Hilbert C��modules of interest will be constructed in the following way Recall that a
pre�C��algebra is a ��algebra satisfying all properties of a C��algebra except perhaps completenessGiven a pre�C��algebra !B� de�ne a pre�Hilbert !B�module !E � !B as in De�nition ���� exceptthat the �nal completeness condition is omitted
Proposition ��� In a pre�Hilbert !B�module and hence in a Hilbert B�module one has theinequalities
k �B k � k � k k B k� ��� �
h���iBh���iB � k � k� h���iB� �����
k h���iB k � k � k k � k � �����
To prove ��� � one uses ������ ������ ��� �� and ��� � For ����� we substitute �h���iB��for � in the inequality h���iB � � Expanding� the �rst term equals h���iBh���iBh���iBThen use ������ and replace � by �� k � k The inequality ����� is immediate from ����� �
Corollary ��� A pre�Hilbert !B�module !E � !B can be completed to a Hilbert B�module�
One �rst completes !E in the norm ������ obtaining E Using ��� �� the !B�action on !E extendsto a B�action on E The completeness of B and ����� then allow one to extend the !B�valuedsesquilinear form on !E to a B�valued one on E It is easily checked that the required propertieshold by continuity �
� � HILBERT C��MODULES AND INDUCED REPRESENTATIONS
In Example ���� it is almost trivial to see that A and H are the closures of !A �de�ned over!A� and of a dense subspace D� respectively
A Hilbert C��module E � B de�nes a certain C��algebra C��E �B�� which plays an importantrole in the induction theory in �� A map A � E � E for which there exists a map A� � E � Esuch that
h�� A�iB � hA����iB �����
for all ��� � E is called adjointable
Theorem ��� An adjointable map is automatically C �linear B�linear that is �A��B � A��B�for all � � E and B � B and bounded� The adjoint of an adjointable map is unique and themap A� A� de�nes an involution on the space C��E �B� of all adjointable maps on E�
Equipped with this involution and with the norm ��� de�ned with respect to the norm ����on E the space C��E �B� is a C��algebra�
Each element A � C��E �B� satis�es the bound
hA�� A�iB �k A k� h���iB �����
for all � � E� The de�ning action of C��E �B� on E is non�degenerate� We write C��E �B� �E � B�
The property of C �linearity is immediate To establish B�linearity one uses ������ this alsoshows that A� � C��E �B� when A � C��E �B�
To prove boundedness of a given adjointable map A� �x � � E and de�ne T� � E � B byT�� �� hA�A���iB It is clear from ����� that k T� k�k A�A� k� so that T� is bounded Onthe other hand� since A is adjointable� one has T�� � h�� A�A�iB� so that� using ����� onceagain� one has k T�� k�k A�A� k k � k Hence supfk T� k j k � k� �g � � by the principleof uniform boundedness �here it is essential that E is complete� It then follows from ����� thatk A k��
Uniqueness and involutivity of the adjoint are proved as for Hilbert spaces� the former followsfrom ������ the latter in addition requires ����
The space C��E �B� is norm�closed� as one easily veri�es from ����� and ����� that if An � Athen A�n converges to some element� which is precisely A� As a norm�closed space of linear mapson a Banach space� C��E �B� is a Banach algebra� so that its satis�es ����� To check ��� � oneinfers from ����� and the de�nition ����� of the adjoint that k A k��k A�A k� then use Lemma����
Finally� it follows from ������ ������ and ����� that for �xed � � E the map A � h�� A�iBfrom C��E �B� to B is positive Replacing A by A�A in ����� and using ��� � and ����� thenleads to �����
To prove the �nal claim� we note that� for �xed ��� � E � the map Z � �h�� ZiB is inC��E �B� When the right�hand side vanishes for all ��� it must be that h�� ZiB � � for all ��hence for � � Z� so that Z � � Here we used the fact that �B � � for all � and B in the linearspan of hE � EiB implies B � �� for by ����� it implies that h���iBB � � �
Under a further assumption �which is by no means always met in our examples� one cancompletely characterize C��E �B� A Hilbert C��module over B is called self�dual when everybounded B�linear map � � E � B is of the form ���� � h���iB for some � � E
Proposition ��� In a self�dual Hilbert C��module E � B the C��algebra C��E �B� coincideswith the space L�E�B of all bounded C �linear and B�linear maps on E�
In view of Theorem ��� we only need to show that a given map A � L�E�B is adjointableIndeed� for �xed � � E de�ne �A�� � E � B by �A���Z� �� h�� AZiB By self�duality this mustequal h�� ZiB for some �� which by de�nition is A�� �
In the context of Example ����� one may wonder what C��A�A� is The map � � A� B�A�given by ����� is easily seen to map A into C��A�A� This map is isometric �hence injective�
��� The C��algebra of a Hilbert C��module �
Using ������ one infers that A��B� � ��AB� for all A�B � A Hence ��A� is an ideal in C��A�A�When A has a unit� one therefore has C��A�A� � ��A� � A� cf the proof of ���
When A has no unit� C��A�A� is the so�called multiplier algebra of A One may computethis object by taking a faithful non�degenerate representation � � A� B�H�� it can be shown thatC��A�A� is isomorphic to the idealizer of ��A� in B�H� �this is the set of all B � B�H� for whichB��A� � ��A� for all A � A� One thus obtains
C��C��X�� C��X�� � Cb�X�� �����
C��B��H��B��H�� � B�H�� �����
Eq ����� follows by taking ��C��X�� to be the representation on L��X� by multiplication opera�tors �where L� is de�ned by a measure with support X�� and ����� is obtained by taking ��B��H��to be the de�ning representation� see the paragraph following ����
In Example ���� the C��algebra C��H� C � coincides with B�H�� because every boundedoperator has an adjoint Its subalgebra B��H� of compact operators has an analogue in thegeneral setting of Hilbert C��modules as well
��� The C�algebra of a Hilbert C�module
In preparation for the imprimitivity theorem� and also as a matter of independent interest� weintroduce the analogue for Hilbert C��modules of the C��algebra B��H� of compact operators ona Hilbert space This is the C��algebra most canonically associated to a Hilbert C��module
Denition ��� The C��algebra C�� �E �B� of �compact�operators on a Hilbert C��module E � B
is the C��subalgebra of C��E �B� generated by the adjointable maps of the type TB�� where ��� �E and
TB�� Z �� �h�� ZiB� �����
We write C�� �E �B�� E � B and call this a dual pair�
The word %compact& appears between quotation marks because in general elements of C�� �E �B�need not be compact operators The signi�cance of the notation introduced at the end of thede�nition will emerge from Theorem ��� below Using the �trivially proved� properties
�TB�� �� � TB ��� �����
ATB�� � TBA�� � �����
TB�� A � TB��A� � ��� �
where A � C��E �B�� one veri�es without di�culty that C�� �E �B� is a �closed ��sided� ideal inC��E �B�� so that it is a C��algebra by Theorem ��� From ��� � and ����� one �nds the bound
k TB�� k�k � k k � k � �����
One sees from the �nal part of the proof of Theorem ��� that C�� �E �B� acts non�degenerately onE
When C�� �E �B� has a unit it must coincide with C��E �B�
Proposition ��� �� When E � B � A see Example ������� one has
C�� �A�A� � A� �����
This leads to the dual pair A� A� A�
�� For E � H and B � C see Example ������� one obtains
C�� �H� C � � B��H�� �����
whence the dual pair B��H�� H� C �
� � HILBERT C��MODULES AND INDUCED REPRESENTATIONS
One has TA�� � ������� see ����� Since � � A � B�A� is an isometric morphism� the map
� from the linear span of all TA�� to A� de�ned by linear extension of ��TA�� � � ���� is anisometric morphism as well It is� in particular� injective When A has a unit it is obvious that� is surjective� in the non�unital case the existence of an approximate unit implies that the linearspan of all ��� is dense in A Extending � to C�� �A�A� by continuity� one sees from Corollary��� that ��C�� �A�A�� � A
Eq ����� follows from De�nition ���� and the fact that the linear span of all T C
�� isBf �H� �
A Hilbert C��module E over B is called full when the collection fh���iBg� where ��� runover E � is dense in B A similar de�nition applies to pre�Hilbert C��modules
Given a complex linear space E � the conjugate space E is equal to E as a real vector space� buthas the conjugate action of complex scalars
Theorem ��� Let E be a full Hilbert B�module� The expression
h���iC���E�B �� TB�� �����
in combination with the right�action �R�A�� �� A�� where A � C�� �E �B� de�nes E as a fullHilbert C��module over C�� �E �B�� In other words from E � B one obtains E � C�� �E �B�� Theleft�action �L�B�� �� �B� of B on E implements the isomorphism
C�� �E � C�� �E �B�� � B� �����
We call A �� C�� �E �B�� in the references to ���� etc below one should substitute A for B whenappropriate The properties ����� ������ and ����� follow from ������ ��� �� and Lemma ����respectively
To prove ������ we use ����� with � � �� ����� with Z � �� ������ ������ and ����� toshow that h���iA � � implies k h���i�
Bk� � Since h���iB is positive by ������ this implies
h���iB � �� hence � � � by �����It follows from ����� and ��� � that each �L�B� is adjointable with respect to h � iA Moreover�
applying ������ ������ ������ and ��� � one �nds that �L�B� is a bounded operator on E withrespect to k � kA� whose norm is majorized by the norm of B in B The map �L is injective becauseE is non�degenerate as a right�B�module
Let Ec be the completion of E in k � kA� we will shortly prove that Ec � E It follows fromthe previous paragraph that �L�B� extends to an operator on Ec �denoted by the same symbol��and that �L maps B into C��Ec�A� It is trivial from its de�nition that �L is a morphism Nowobserve that
�L�h���iB� � TA�� � �����
for the de�nitions in question imply that
TA�� Z � �h�� ZiA � TBZ� � � Zh���iB� �����
The fullness of E � B and the de�nition of C�� �Ec�A� imply that �L � B � C�� �Ec�A� is anisomorphism In particular� it is norm�preserving by Lemma ��
The space E is equipped with two norms by applying ����� with B or with A� we write k � kBand k � kA From ����� and ����� one derives
k � kA�k � kB � �����
For � � E we now use ������ the isometric nature of �L� and ����� to �nd that
k � kB�k TA��� k�
� � �����
From ����� withB� A one then derives the converse inequality to ������ so that k � kA�k � kBHence Ec � E � as E is complete in k � kB by assumption The completeness of E as a Hilbert B�module is equivalent to the completeness of E as a Hilbert A�module
��� Morita equivalence �
We have now proved ����� Finally noticing that as a Hilbert C��module over A the space Eis full by de�nition of C�� �E �B�� the proof of Theorem ��� is ready �
For later reference we record the remarkable identity
h���iC���E�BZ � �h�� ZiB� ��� �
which is a restatement of �����
��� Morita equivalence
The imprimitivity theorem establishes an isomorphism between the respective representation the�ories of two C��algebras that stand in a certain equivalence relation to each other
Denition ��� Two C��algebras A and B are Morita�equivalent when there exists a full
Hilbert C��module E � B under which A � C�� �E �B�� We write AM B and A� E � B�
Proposition ��� Morita equivalence is an equivalence relation in the class of all C��algebras�
The re#exivity property BM B follows from ������ which establishes the dual pair B� B�
B Symmetry is implied by ������ proving that A� E � B implies B� E � A
The proof of transitivity is more involved When AM B and B
M C we have the chain of dualpairs
A� E� � B� E� � C�
We then form the linear space E��B E� �which is the quotient of E��E� by the ideal IB generatedby all vectors of the form ��B ��� ��� �B���� which carries a right�action �
R�C� given by
�R
�C���� �B ��� �� �� �B ���C�� �����
Moreover� we can de�ne a sesquilinear map h � iC
on E� �B E� by
h�� �B ����� �B ��iC �� h��� h�����iB��iC� �����
With ����� this satis�es ���� and ������ as explained prior to ������ one may therefore constructa Hilbert C��module� denoted by E � C �Remarkably� if one looks at ����� as de�ned onE� � E�� the null space of ����� is easily seen to contain IB� but in fact coincides with it� so thatin constructing E one only needs to complete E� �B E��
Apart from the right�action �R
�C�� the space E carries a left�action �L �A�� the operator
�L
�A���� �B ��� �� �A����B �� �����
is bounded on E� �B E� and extends to E We now claim that
C�� �E�C� � �L �A�� �����
Using ������ the de�nition of �B� and ������ it is easily shown that
�L
�TB��h��� �iB� ��(� �B (� � �� �B h�����h���(�iBiB(�� �����
Now use the assumption C�� �E��C� � B� as in ������ with B � C� and E � E�� this yieldsh���iB � TC�� Substituting this in the right�hand side of ������ and using ����� with B� C�the right�hand side of ����� becomes �� �B ��h��h���(�iB�(�iC Using �B� � �L�B�� �see����� ����� with B� C� ������ and ����� with B� C� we eventually obtain
TC��B��� �B � � �L
�TB��h��� �iB� ��� �����
This leads to the inclusion C�� �E�C� �L �A� To prove the opposite inclusion� one picks a
double sequence f�i���
i�g such that
PNi TC
�i�� i�
is an approximate unit in B � C�� �E��C� One
� � HILBERT C��MODULES AND INDUCED REPRESENTATIONS
has limN
PNi �i
�h�i�� ZiC � Z from ������ and a short computation using ����� with ����� then
yields
limN
NXi
TC��B�i�� �B i�
� �L
�TB��� ���
Hence �L �A� C�� �E�C�� and combining both inclusions one �nds �����
Therefore� one has the dual pair A � E � C� implying that AM C This proves transitiv�
ity �
Here is a simple example of this concept
Proposition ��� The C��algebra B��H� of compact operators is Morita�equivalent to C withdual pair B��H�� H� C � In particular the matrix algebra Mn�C � is Morita�equivalent to C �
This is immediate from ����� In the �nite�dimensional case one has Mn�C � � C n � C � whereMn�C � and C act on C n in the usual way The double Hilbert C��module structure is completedby specifying
hz� wiC � ziwi�
�hz� wiMn�C �ij � ziwj � �����
from which one easily veri�es ��� � �
Since Mn�C �M C and C
M Mm�C �� one has Mn�C �M Mm�C � This equivalence is imple�
mented by the dual pair Mn�C � �Mn�m�C � �Mm�C �� where Mn�m�C � is the space of complexmatrices with n rows and m columns We leave the details as an exercise
In practice the following way to construct dual pairs� and therefore Morita equivalences� isuseful
Proposition ��� Suppose one has
� two pre�C��algebras !A and !B�
� a full pre�Hilbert !B�module !E�
� a left�action of !A on !E such that !E can be made into a full pre�Hilbert !A�module with respectto the right�action �R�A�� �� A���
� the identityh���i�AZ � �h�� Zi �B �����
for all ���� Z � !E relating the two Hilbert C��module structures�
� the bounds
h�B��Bi�A � k B k� h���i�A� �����
hA�� A�i �B � k A k� h���i �B ��� �
for all A � !A and B � !B�
Then AM B with dual pair A� E � B where E is the completion of !E as a Hilbert B�module�
Using Corollary ��� we �rst complete !E to a Hilbert B�module E By ��� �� which impliesk A� k�k A k k � k for all A � !A and � � !E � the action of !A on !E extends to an action of A on
E Similarly� we complete !E to a Hilbert A�module Ec� by ����� the left�action �L�B�� �� �B�
extends to an action of B on Ec As in the proof of Theorem ���� one derives ����� and its
converse for � � !E � so that the B�completion E of !E coincides with the A�completion Ec of !E � thatis� Ec � E
��� Rie�el induction �
Since E is a full pre�Hilbert !A�module� the A�action on E is injective� hence faithful It followsfrom ������ Theorem ���� and �once again� the fullness of E � that A � C�� �E �B� In particular�each A � A automatically satis�es ����� �
Clearly� ����� is inspired by ��� �� into which it is turned after use of this proposition Wewill repeatedly use Proposition ��� in what follows� see ��� and ����
��� Rie�el induction
To formulate and prove the imprimitivity theorem we need a basic technique� which is of interestalso in a more general context Given a Hilbert B�module E � the goal of the Rie�el inductionprocedure described in this section is to construct a representation � of C��E �B� from a repre�sentation � of B In order to explicate that the induction procedure is a generalization of theGNS�construction ���� we �rst induce from a state � on B� rather than from a representation�
Construction ��� Suppose one has a Hilbert C��module E � B�
�� Given a state � on B de�ne the sesquilinear form g� � �
� on E by
������
� �� ��h���iB�� �����
Since � and h � iB are positive cf� ���� this form is positive semi�de�nite� Its null spaceis
!N � f� � E j ������
� � �g� �����
�� The form g� � �
� projects to an inner product g� � �on the quotient E� !N� If !V � E � E� !N
is the canonical projection then by de�nition
�� !V�� !V��
�� ������
� � �����
The Hilbert space !H is the closure of E� !N in this inner product�
�� The representation !��C��E �B�� is �rstly de�ned on E� !N � !H by
��A� !V� �� !VA�� �����
it follows that !� is continuous� Since E� !N is dense in !H the operator !��A� may be
de�ned on all of !H by continuous extension of ���� where it satis�es ������
The GNS�construction ��� is a special case of ���� obtained by choosing E � B � A� asexplained in Example ����
The analogue of ������ of course applies here The continuity of !� follows from ����� and
������ which imply that k !��A� !V� k�� ��A�� A��
� Using ������ ������ and ����� in succes�sion� one �nds that
k !��A� k�k A k � �����
On the other hand� from the proof of Theorem ���� one sees that
k A k�� supfj��A�A�j j� � S�A�g� �����
Applying ����� to B� used with the de�nition of k A k for A � C��E �B�� implies that
k A k� supfk !��A� k� � � S�B�g� �����
A similar argument combined with Corollary ��� shows that !� is faithful �hence norm�preserving�when � is As a corollary� one infers a useful property� which will be used� eg� in the proof ofTheorem ���
� HILBERT C��MODULES AND INDUCED REPRESENTATIONS
Lemma ��� Let A � C��E �B� satisfy h�� A�iB � � for all � � E� Then A � ��
Take a faithful state � on B� the condition implies that !��A� � � �
When one starts from a representation ��B� rather than from a state� the general constructionlooks as follows
Construction ��� Start from a Hilbert C��module E � B�
�� Given a representation ��B� on a Hilbert space H with inner product � � � the sesquilinearform � � �� is de�ned on E �H algebraic tensor product by sesquilinear extension of
��� v��� w�� �� �v� ��h���iB�w�� �����
where v� w � H� This form is positive semi�de�nite because � � � and h � iB are� The nullspace is
N � f!� � E �Hj � !�� !��� � �g� �����
As in ����� we may equally well write
N � f!� � E �Hj � !�� !��� � � � !� � E �Hg� ��� �
�� The form � � �� projects to an inner product � � � on the quotient E �H�N de�ned by
�V !�� V !�� �� � !�� !��� � �����
where V � E � H � E � H�N is the canonical projection� The Hilbert space H is theclosure of E �H�N in this inner product�
�� The representation ��C��E �B�� is then de�ned on H by continuous extension of
��A�V !� �� V�A� I!��� �����
where I is the unit operator on H� the extension in question is possible since
k ��A� k�k A k � �����
To prove that the form de�ned in ����� is positive semi�de�nite� we assume that ��B� is cyclic
�if not� the argument below is repeated for each cyclic summand� see ���� With !� �P
i �iviand vi � ��Bi�( �where ( is a cyclic vector for ��B��� one then uses ������ ������ and �����
to �nd � !�� !��� � �v� ��h���iB�v� with � ��P
i �iBi Hence � !�� !��� � � by ����� and thepositivity of � � B� B�H�
Similarly� one computes k ��A�V !� k�� �v� ��hA�� A�iB�v� from ����� and ������ ac�cording to ����� and the property k ��A� k�k A k �cf the text after ����� this is bounded by
k A k� �v� ��h���iB�v� Since the second factor equals k V !� k�� this proves ����� �
Paraphrasing the comment after the �rst version of the construction� � is faithful when � isAlso� it is not di�cult to verify that � is non�degenerate when � is
To interrelate the above two formulations� one assumes that � is cyclic� with cyclic vector (
Then de�ne a linear map !U � E � E �H by
!U� �� �� (� �� ��
According to ������ ������ and ���� �� this map has the property
� !U�� !U��� � �� !�� !��
� � �� ��
By ����� and ����� the map !U therefore quotients to a unitary isomorphism U � !H � H� whichby ����� and ����� duly intertwines !� and �
Of course� any subspace of C��E �B� may be subjected to the induced representation � Thisparticularly applies when one has a given �pre�� C��algebra A and a ��homomorphism � � A �C��E �B�� leading to the representation ��A� on H Further to an earlier comment� one veri�esthat � is non�degenerate when � and � are With slight abuse of notation we will write ��A�for ����A�� The situation is depicted in Figure �
��� The imprimitivity theorem �
A E B H
H
�
�
HHHHHHHHHj
� �
induction
� �
�
Figure �� Rie�el induction
A E B H
B E A H �� H�
� � �
� � �
�����R
�
�
B H�
�����R��
�
Figure �� Quantum imprimitivity theorem� H� � H and �� � �
�� The imprimitivity theorem
After this preparation� we pass to the imprimitivity theorem
Theorem ��� There is a bijective correspondence between the non�degenerate representationsof Morita�equivalent C��algebras A and B preserving direct sums and irreducibility� This corre�spondence is as follows�
Let the pertinent dual pair be A� E � B� When ���A� is a representation on a Hilbert spaceH� there exists a representation ��B� on a Hilbert space H such that �� is equivalent to theRie�el�induced representation � de�ned by ���� and the above dual pair�
In the opposite direction a given representation ��B� is equivalent to the Rie�el�inducedrepresentation �� de�ned with respect to some representation ���A� and the dual pair B� E � A�
Taking ���A� � ��A� as just de�ned one has ���B� � ��B�� Conversely taking ��B� ����B� one has ��A� � ���A��
See Figure � Starting with ��B�� we construct ��A� with Rie�el induction from the dualpair A � E � B� relabel this representation as ���A�� and move on to construct ���B� fromRie�el induction with respect to the dual pair B � E � A We then construct a unitary mapU � H� � H which intertwines �� and �
We �rst de�ne !U � E � E �H � H by linear extension of
!U�� �� v �� ��h���iB�v� �� ��
� � HILBERT C��MODULES AND INDUCED REPRESENTATIONS
Note that !U is indeed C �linear Using �� ��� the properties ����� and ����� with �� �� ������and ������ one obtains
� !U�� ��� � v�� !U�� ��� � v�� � ��� � v�� TB
������� � v��� � �� ��
Now use the assumption A � C�� �E �B� to use ��� �� and subsequently ����� and ������ allread from right to left The right�hand side of �� �� is then seen to be equal to �V�� �v�� �
�h�����iA�V�� � v�� Now put � � �� and H � H� � and use ����� and �����from right to left� with �� This shows that
� !U�� ��� � v�� !U�� ��� � v�� � �V���� � V�� � v��� V���� � V�� � v���� � �� ��
In particular� !U annihilates �� !�� where !� � E �H� whenever !� � N or ��V !� � N� Hencewe see from the construction �rstly of H � H� from E � H� and secondly of H� from E � H�
�cf ����� that !U descends to an isometry U � H� � H� de�ned by linear extension of
UV���� V�� v� �� !U�� �� v � ��h���iB�v� �� ��
Using the assumptions that the Hilbert C��module E � B is full and that the representation��B� is non�degenerate� we see that the range of !U and hence of U is dense in H� so that U isunitary
To verify that U intertwines �� and �� we use �� �� and ������ with �� � to compute
U���B�V���� V�� v� � ��h�L�B����iB�v� �� �
where the left�action of B � B on � � E is as de�ned in ��� Thus writing �L�B�� � �B��using ������ ����� with �� �� and �� �� from right to left� the right�hand side of �� � is seento be ��B�UV���� V�� v� Hence U���B� � ��B�U for all B � B
Using the proof that the Morita equivalence relation is symmetric �see ����� one immediatelysees that the construction works in the opposite direction as well
It is easy to verify that � � �� � �� leads to � � �� � �
�
This also proves that thebijective correspondence ��B� � ��A� preserves irreducibility� when � is irreducible and �
isn�t� one puts � � �� as above� decomposes �� � ��� � ��� � then decomposes the inducedrepresentation ���B� as �� � ��
� � ���
� and thus arrives at a contradiction� since �� � � �
Combined with Proposition ���� this theorem leads to a new proof of Corollary �����
��� Group C�algebras
In many interesting applications� and also in the theory of induced representation as originallyformulated for groups by Frobenius and Mackey� the C��algebra B featuring in the de�nition of aHilbert C��module and in Rie�el induction is a so�called group C��algebra
We start with the de�nition of the group algebra C��G� of a �nite group with n�G� elements�one then usually writes C �G� instead of C��G� As a vector space� C �G� consist of all complex�valued functions on G� so that C �G� � C n�G This is made into a ��algebra by the convolution
f � g�x� ��X
y�z�Gjyz�x
f�y�g�z� �� ��
and the involution
f��x� �� f�x���� �� ��
It is easy to check that the multiplication � is associative as a consequence of the associativity ofthe product in G In similar vein� the operation de�ned by �� �� is an involution because of theproperties �x����� � x and �xy��� � y��x�� at the group level
A representation � of C �G� on a Hilbert space H is de�ned as a morphism � � C �G� � B�H�
�� Group C��algebras �
Proposition ��� There is a bijective correspondence between non�degenerate representations �of the ��algebra C �G� and unitary representations U of G which preserves unitary equivalence anddirect sums and therefore preserves irreducibility� This correspondence is given in one directionby
��f� ��Xx�G
f�x�U�x�� �� ��
and in the other byU�x� �� ���x�� �����
where �x�y� �� ��xy��
It is elementary to verify that � is indeed a representation of C �G� when U is a unitaryrepresentation of G� and vice versa Putting x � e in ����� yields ���e� � I� so that � cannot bedegenerate
When U��x� � V U��x�V � for all x � G then evidently ���f� � V ���f�V � for all f � C �G�The converse follows by choosing f � �x Similarly� ��f� � ���f� � ���f� for all f i� U�x� �U��x�� U��x� for all x �
We can de�ne a C��norm on C �G� by taking any faithful representation �� and putting k f k��k��f� k Since C �G� is a �nite�dimensional vector space it is complete in this norm� which thereforeis independent of the choice of � by Corollary ���
Let now G be an arbitrary locally compact group �such as a �nite�dimensional Lie group� Wealso assume that G is unimodular� that is� each left Haar measure is also right�invariant Thisassumption is not necessary� but simpli�es most of the formulae We denote Haar measure by dx�it is unique up to normalization Unimodularity implies that the Haar measure is invariant underinversion x � x�� When G is compact we choose the normalization so that
RG dx � � The
Banach space L��G� and the Hilbert space L��G� are de�ned with respect to the Haar measureThe convolution product is de�ned� initially on Cc�G�� by
f � g�x� ��
ZG
dy f�xy���g�y�� �����
it is evident that for a �nite group this expression specializes to �� �� The involution is given by�� �� As in the �nite case� one checks that these operations make C c �G� a ��algebra� this timeone needs the invariance of the Haar measure at various steps of the proof
Proposition ��� The operations ���� and ���� are continuous in the L��norm� one has
k f � g k� � k f k� k g k�� �����
k f� k� � k f k� � �����
Hence L��G� is a Banach ��algebra under the continuous extensions of ���� and ���� fromCc�G� to L��G��
Recall the de�nition of a Banach ��algebra below ����It is obvious from invariance of the Haar measure under x� x�� that k f� k��k f k�� so that
the involution is certainly continuous The proof of ����� is a straightforward generalization ofthe case G � R� cf ��� � This time we have
k f � g k��ZG
dx jZG
dy f�xy���g�y�j �ZG
dy jg�y�jZG
dx jf�xy���j
�
ZG
dy jg�y�jZG
dx jf�x�j �k f k� k g k��
which is ����� �
In order to equip L��G� with a C��norm� we construct a faithful representation on a Hilbertspace
�� � HILBERT C��MODULES AND INDUCED REPRESENTATIONS
Proposition ��� For f � L��G� the operator �L�f� on L��G� de�ned by
�L�f�� �� f ��� �����
is bounded satisfying k �L�f� k�k f k�� The linear map �L � L��G� � B�L��G�� is a faithfulrepresentation of L��G� seen as a Banach ��algebra as in ������
Introducing the left�regular representation UL of G on L��G� by
UL�y���x� �� ��y��x�� �����
it follows that
�L�f� �
ZG
dx f�x�UL�x�� ��� �
The boundedness of �L�f� then follows from Lemma ��� below One easily veri�es that �L�f�g� ��L�f��L�g� and �L�f�� � �L�f��
To prove that �L is faithful� we �rst show that L��G� possesses the analogue of an approximateunit �see ��� for C��algebras� When G is �nite� the delta�function �e is a unit in C �G� Forgeneral locally compact groups one would like to take the Dirac ���function� as a unit� but thisdistribution is not in L��G�
Lemma ��� The Banach ��algebra L��G� has an approximate unit I� in the sense that ����� ���� hold for all A � L��G� and k � k�k � k��
Pick a basis of neighbourhoods N� of e� so that each N� is invariant under x� x��� this basis ispartially ordered by inclusion Take I� � N��N�
� which is the characteristic function of N� timesa normalization factor ensuring that k I� k�� � Eq ����� then holds by virtue of �� �� and theinvariance of N� under inversion By construction� the inequality ����� holds as an equality Onehas I�� f�x� � N�
RN�
dy f�y��x� and f �I��x� � N�
RN�
dy f�xy�� For f � Cc�G� one therefore
has lim� I� � f � f and lim� f � I� � f pointwise �ie� for �xed x� The Lebesgue dominatedconvergence theorem then leads to ����� for all A � Cc�G�� and therefore for all A � L��G�� sinceCc�G� is dense in L��G� �
To �nish the proof of ���� we now note from ����� that �L�f� � � implies f �� � � for all� � L��G�� and hence certainly for � � I� Hence k f k�� � by Lemma ���� so that f � � and�L is injective �
Denition ��� The reduced group C��algebra C�r �G� is the smallest C��algebra inB�L��G��containing �L�Cc�G��� In other words C�r �G� is the closure of the latter in the norm
k f kr��k �L�f� k � �����
Perhaps the simplest example of a reduced group algebra is obtained by taking G � R
Proposition ��� One has the isomorphism
C�r �R� � C��R�� �����
It follows from the discussion preceding �� that the Fourier transform ����� maps L��G�into a subspace of C��R� which separates points on R It is clear that for every p � R there is an
f � L��R� for which �f�p� � � In order to apply Lemma ���� we need to verify that
k f kr�k �f k� � �����
Since the Fourier transform turns convolution into pointwise multiplication� the left�regular repre�sentation �L on L��R� is Fourier�transformed into the action on L��R� by multiplication operatorsHence ����� follows from Lemma ��� �
�� Group C��algebras ��
This example generalizes to arbitrary locally compact abelian groups Let �G be the set of allirreducible unitary representations U� of G� such representations are necessarily one�dimensional�
so that �G is nothing but the set of characters on G The generalized Fourier transform �f off � L��G� is a function on �G� de�ned as
�f�� ��
ZG
dx f�x�U��x�� �����
By the same arguments as for G � R� one obtains
C�r �G� � C�� �G�� �����
We return to the general case� where G is not necessarily abelian We have now found a C��algebra which may play the role of C �G� for locally compact groups Unfortunately� the analogueof Proposition ��� only holds for a limited class of groups Hence we need a di�erent constructionLet us agree that here and in what follows� a unitary representation of a topological group is alwaysmeant to be continuous
Lemma ��� Let U be an arbitrary unitary representation of G on a Hilbert space H� Then��f� de�ned by
��f� ��
ZG
dx f�x�U�x� �����
is bounded withk ��f� k�k f k� � �����
The integral ����� is most simply de�ned weakly� that is� by its matrix elements
��� ��f��� ��
ZG
dx f�x���� U�x����
Since U is unitary� we have j��� ��f���j � �F� F �L��G for all � � H� where F �x� �� k � kpjf�x�jThe Cauchy�Schwarz inequality then leads to j��� ��f���j � k f k�k � k� Lemma ���� thenleads to ����� �
Alternatively� one may de�ne ����� as a Bochner integral We explain this notion in a moregeneral context
Denition ��� Let X be a measure space and let B be a Banach space� A function f � X � Bis Bochner�integrable with respect to a measure � on X i�
� f is weakly measurable that is for each functional � � B� the function x � ��f�x�� ismeasurable�
� there is a null set X� � X such that ff�x�jx � XnX�g is separable�
� the function de�ned by x�k f�x� k is integrable�
It will always be directly clear from this whether a given operator� or vector�valued integralmay be read as a Bochner integral� if not� it is understood as a weak integral� in a sense alwaysobvious from the context The Bochner integral
RXd��x�f�x� can be manipulated as if it were an
ordinary �Lebesgue� integral For example� one has
kZX
d��x� f�x� k�ZX
d��x� k f�x� k � �����
Thus reading ����� as a Bochner integral� ����� is immediate from �����The following result generalizes the correspondence between UL in ����� and �L in ��� � to
arbitrary representations
�� � HILBERT C��MODULES AND INDUCED REPRESENTATIONS
Theorem ��� There is a bijective correspondence between non�degenerate representations � ofthe Banach ��algebra L��G� which satisfy ���� and unitary representations U of G� This corre�spondence is given in one direction by ���� and in the other by
U�x���f�( �� ��fx�(� �����
where fx�y� �� f�x��y�� This bijection preserves direct sums and therefore irreducibility�
Recall from ��� that any non�degenerate representation of a C��algebra is a direct sum ofcyclic representations� the proof also applies to L��G� Thus ( in ����� stands for a cyclic vectorof a certain cyclic summand of H� and ����� de�nes U on a dense subspace of this summand� itwill be shown that U is unitary� so that it can be extended to all of H by continuity
Given U � it follows from easy calculations that ��f� in ����� indeed de�nes a representationIt is bounded by Lemma ��� The proof of non�degeneracy makes use of Lemma ��� Since �is continuous� one has lim� ��I�� � I strongly� proving that � must be non�degenerate
To go in the opposite direction we use the approximate unit once more� it follows from ������from which the continuity of U is obvious� that U�x���f�( � lim� ��Ix����f�( Hence U�x� �lim� ��Ix�� strongly on a dense domain The property U�x�U�y� � U�xy� then follows from �����and ����� The unitarity of each U�x� follows by direct calculation� or from the following argumentSince k ��Ix�� k�k Ix� k�� �� we infer that k U�x� k� � for all x Hence also k U�x��� k� �� whichis the same as k U�x��� k� � We see that U�x� and U�x��� are both contractions� this is onlypossible when U�x� is unitary
Finally� if U is reducible there is a projection E such that �E�U�x� � � for all x � G It followsfrom ����� that ���f�� E � � for all f � hence � is reducible Conversely� if � is reducible then�E� ��Ix�� � � for all x � G� by the previous paragraph this implies �E�U�x� � � for all x The�nal claim then follows from Schur�s lemma ����� �
This theorem suggests looking at a di�erent object from C�r �G� Inspired by ���� one puts
Denition ���� The group C��algebra C��G� is the closure of the Banach ��algebra algebraL��G� in the norm
k f k��k �u�f� k� ��� �
where �u is the direct sum of all non�degenerate representations � of L��G� which are bounded asin �����
Equivalently C��G� is the closure of L��G� in the norm
k f k�� sup�k ��f� k� �����
where the sum is over all representations ��L��G�� of the form ���� in which U is an irre�ducible unitary representation of G and only one representative of each equivalence class of suchrepresentations is included�
The equivalence between the two de�nitions follows from ������ and Theorem ���
Theorem ���� There is a bijective correspondence between non�degenerate representations �of the C��algebra C��G� and unitary representations U of G given by continuous extension of���� and ����� This correspondence preserves irreducibility�
It is obvious from ������ and ��� � that for any representation ��C��G�� and f � L��G� onehas
k ��f� k�k f k�k f k� � �����
Hence the restriction ��L��G�� satis�es ������ and therefore corresponds to U�G� by Theorem��� Conversely� given U�G� one �nds ��L��G�� satisfying ����� by ���� it then follows from����� that one may extend � to a representation of C��G� by continuity �
�� C��dynamical systems and crossed products ��
In conjunction with ������ the second de�nition of C��G� stated in ���� implies that forabelian groups C��G� always coincides with C�r �G� The reason is that for � �G one has ���f� ��f�� � C � so that the norms ����� and ����� coincide In particular� one has
C��Rn � � C��Rn �� �����
For general locally compact groups� looking at ��� we see that
C�r �G� � �L�C��G�� � C��G�� ker��L�� �����
A Lie group group is said to be amenable when the equality C�r �G� � C��G� holds� in otherwords� �L�C��G�� is faithful i� G is amenable We have just seen that all locally compact abeliangroups are amenable It follows from the Peter�Weyl theorem that all compact groups are amenableas well However� non�compact semi�simple Lie groups are not amenable
��� C�dynamical systems and crossed products
An automorphism of a C��algebra A is an isomorphism between A and A It follows fromDe�nitions ��� and ��� that ���A�� � �A� for any A � A any automorphism �� hence
k ��A� k�k A k �����
by �����One A has a unit� one has
��I� � I �����
by ����� and the uniqueness of the unit When A has no unit� one may extend � to an automor�phism �I of the unitization AI by
�I�A " �I� �� ��A� " �I� �����
Denition ��� An automorphic action � of a group G on a C��algebra A is a group homo�morphism x� �x such that each �x is an automorphism of A� In other words one has
�x � �y�A� � �xy�A�� �����
�x�AB� � �x�A��x�A�� �����
��A�� � ��A�� ��� �
for all x� y � G and A�B � A�A C��dynamical system �G�A� �� consists of a locally compact group G a C��algebra A and
an automorphic action of G on A such that for each A � A the function from G to A de�ned byx�k �x�A� k is continuous�
The term �dynamical system� comes from the example G � R and A � C��S�� where R acts on Sby t � � �t�� and �t�f� � � �t� Hence a general C��dynamical system is a non�commutativeanalogue of a dynamical system
Proposition ��� Let �G�A� �� be a C��dynamical system and de�ne L��G�A� �� as the spaceof all measurable functions f � G� A for which
k f k���ZG
dx k f�x� k �����
is �nite� The operations
f � g�x� ��
ZG
dy f�y��y�g�y��x��� �����
f��x� �� �x�f�x����� �����
turn L��G�A� �� into a Banach ��algebra�
�� � HILBERT C��MODULES AND INDUCED REPRESENTATIONS
As usual� we have assumed that G is unimodular� with a slight modi�cation one may extendthese formulae to the non�unimodular case The integral ����� is de�ned as a Bochner integral�the assumptions in De�nition ��� are satis�ed as a consequence of the continuity assumption inthe de�nition of a C��dynamical system To verify the properties ����� and ����� one follows thesame derivation as for L��G�� using ����� and ����� The completeness of L��G�A� �� is provedas in the case A � C � for which L��G�A� �� � L��G� �
In order to generalize Theorem ���� we need
Denition ��� A covariant representation of a C��dynamical system �G�A� �� consists ofa pair �U� !�� where U is a unitary representation of G and !� is a non�degenerate representationof A which for all x � G and A � A satis�es
U�x�!��A�U�x�� � !���x�A��� ������
Here is an elegant and useful method to construct covariant representations
Proposition ��� Let �G�A� �� be a C��dynamical system and suppose one has a state � on Awhich is G�invariant in the sense that
���x�A�� � ��A� ������
for all x � G and A � A� Consider the GNS�representation ���A� on a Hilbert space H� withcyclic vector (�� For x � G de�ne an operator U�x� on the dense subspace ���A�(� of H� by
U�x����A�(� �� ����x�A��(� � ������
This operator is well de�ned and de�nes a unitary representation of G on H��
If ���A�(� � ���B�(� then ���A�B���A�B�� � � by ������ Hence ���x�A�B���x�A�B�� � � by ������� so that k ����x�A � B��(� k�� � by ������ Hence ����x�A��(� �����x�B��� so that U�x����A�(� � U�x����B�(�
Furthermore� ����� implies that U�x�U�y� � U�xy�� whereas ������ and ������ imply that
�U�x����A�(� � U�x����B�(�� � ����A�(� � ���B�(���
This shows �rstly that U�x� is bounded on ���A�(�� so that it may be extended to H� bycontinuity Secondly� U�x� is a partial isometry� which is unitary from H� to the closure ofU�x�H� Taking A � �x���B� in ������� one sees that U�x�H� � ���A�(�� whose closure is H�
because �� is cyclic Hence U�x� is unitary �
Note that ������ with ����� or ����� implies that
U�x�(� � (�� ������
Proposition ��� describes the way unitary representations of the Poincar)e group are con�structed in algebraic quantum �eld theory� in which � is then taken to be the vacuum state onthe algebra of local observables of the system in question Note� however� that not all covariantrepresentations of a C��dynamical system arise in this way� a given unitary representation U�G�may may not contain the trivial representation as a subrepresentation� cf ������
In any case� the generalization of Theorem ��� is as follows Recall �����
Theorem ��� Let �G�A� �� be a C��dynamical system� There is a bijective correspondencebetween non�degenerate representations � of the Banach ��algebra L��G�A� �� which satisfy ���� and covariant representations �U�G�� !��A��� This correspondence is given in one direction by
��f� �
ZG
dx !��f�x��U�x�� ������
��� Transformation group C��algebras ��
in the other direction one de�nes Af � x� Af�x� and !�x�f� � y � �x�f�x��y�� and puts
U�x���f�( � ��!�x�f��(� ������
!��A���f�( � ��Af�(� ���� �
where ( is a cyclic vector for a cyclic summand of ��C��G� !A���This bijection preserves direct sums and therefore irreducibility�
The proof of this theorem is analogous to that of ��� The approximate unit in L��G�A� ��is constructed by taking the tensor product of an approximate unit in L��G� and an approximateunit in A The rest of the proof may then essentially be read o� from ��� �
Generalizing ����� we put
Denition ��� Let �G�A� �� be a C��dynamical system� The crossed product C��G�A� �� ofG and A is the closure of the Banach ��algebra algebra L��G�A� �� in the norm
k f k��k �u�f� k� ������
where �u is the direct sum of all non�degenerate representations � of L��G�A� �� which are boundedas in �����
Equivalently C��G�A� �� is the closure of L��G�A� �� in the norm
k f k�� sup�k ��f� k� ������
where the sum is over all representations ��L��G�A� ��� of the form ����� in which �U� !�� is anirreducible covariant representation of �G�A� �� and only one representative of each equivalenceclass of such representations is included�
Here we simply say that a covariant representation �U� !�� is irreducible when the only boundedoperator commuting with all U�x� and !��A� is a multiple of the unit The equivalence betweenthe two de�nitions follows from ������ and Theorem ���
Theorem ��� Let �G�A� �� be a C��dynamical system� There is a bijective correspondencebetween non�degenerate representations � of the crossed product C��G�A� �� and covariant repre�sentations �U�G�� !��A��� This correspondence is given by continuous extension of ����� and����� ������ This correspondence preserves direct sums and therefore irreducibility�
The proof is identical to that of ���� �
�� Transformation group C�algebras
We now come to an important class of crossed products� in which A � C��Q�� where Q is a locallycompact Hausdor� space� and �x is de�ned as follows
Denition ��� A left� action L of a group G on a space Q is a map L � G�Q� Q satisfyingL�e� q� � q and L�x� L�y� q�� � L�xy� q� for all q � Q and x� y � G� If G and Q are locally compactwe assume that L is continuous� If G is a Lie group and Q is a manifold we assume that L issmooth� We write Lx�q� � xq �� L�x� q��
We assume the reader is familiar with this concept� at least at a heuristic level The mainexample we shall consider is the canonical action of G on the coset space G�H �where H is aclosed subgroup of G� This action is given by
x�y H �� �xy H � ������
where �x H �� xH � cf �� etc For example� when G � SO��� and H � SO��� is the subgroupof rotations around the z�axis� one may identify G�H with the unit two�sphere S� in R� TheSO����action ������ is then simply the usual action on R� � restricted to S�
� � HILBERT C��MODULES AND INDUCED REPRESENTATIONS
Assume that Q is a locally compact Hausdor� space� so that one may form the commutativeC��algebra C��Q�� cf �� A G�action on Q leads to an automorphic action of G on C��Q�� givenby
�x� !f� � q � !f�x��q�� ������
Using the fact that G is locally compact� so that e has a basis of compact neighbourhoods� it iseasy to prove that the continuity of the G�action on Q implies that
limx�e
k �x� !f�� !f k� � ������
for all !f � Cc�Q� Since Cc�Q� is dense in C��Q� in the sup�norm� the same is true for !f � C��Q�Hence the function x� �x� !f� from G to C��Q� is continuous at e �as �e� !f� � !f� Using ����� and������ one sees that this function is continuous on all of G Hence �G�C��Q�� �� is a C��dynamicalsystem
It is quite instructive to look at covariant representations �U� !�� of �G�C��Q�� �� in the specialcase that G is a Lie group and Q is a manifold Firstly� given a unitary representation U of a Liegroup G on a Hilbert space H one can construct a representation of the Lie algebra g by
dU�X�� ��d
dtU�Exp�tX���jt��� ������
When H is in�nite�dimensional this de�nes an unbounded operator� which is not de�ned on allof H Eq ������ makes sense when � is a smooth vector for a U � this is an element � � Hfor which the map x � U�x�� from G to H is smooth It can be shown that the set H�
U ofsmooth vectors for U is a dense linear subspace of H� and that the operator idU�X� is essentiallyself�adjoint on H�
U Moreover� on H�U one has
�dU�X�� dU�Y � � dU��X�Y �� ������
Secondly� given a Lie group action one de�nes a linear map X � �X from g to the space of allvector �elds on Q by
�X !f�q� ��d
dt!f�Exp�tX�q�jt��� ������
where Exp � g� G is the usual exponential mapThe meaning of the covariance condition ������ on the pair �U� !�� may now be clari�ed by
re�expressing it in in�nitesimal form For X � g� !f � C�c �Q�� and � � Rnf�g we put
Q��� !X� �� i�dU�X�� ������
Q��� !f� �� !�� !f�� ���� �
From the commutativity of C��Q�� ������� and ������� respectively� we then obtain
i
��Q�
�� !f��Q���!g� � �� ������
i
��Q�
�� !X��Q��� !Y � � Q�
�����X�Y �� ������
i
��Q�
�� !X��Q��� !f� � Q�
���X !f�� ������
These equations hold on the domain H�U � and may be seen as a generalization of the canonical
commutation relations of quantum mechanics To see this� consider the case G � Q � Rn �where the G�action is given by L�x� q� �� q " x If X � Tk is the k�th generator of Rn one has�k �� �Tk � ���qk Taking f � ql� the l�th co�ordinate function on Rn � one therefore obtains�kq
l � �lk The relations ������ � ������ then become
i
��Q�
��qk��Q���ql� � �� ������
i
��Q�
�� !Tk��Q��� !Tl� � �� ������
i
��Q�
�� !Tk��Q���ql� � �lk� ������
��� Transformation group C��algebras ��
Hence one may identify Q���qk� andQ�
�� !Tk� with the quantum position and momentum observables�
respectively �It should be remarked that Q���qk� is an unbounded operator� but one may show
from the representation theory of the Heisenberg group that Q���qk� and Q�
�� !Tk� always possess a
common dense domain on which ������ � ������ are valid�
Denition ��� Let L be a continuous action of a locally compact group on a locally compactspace Q� The transformation group C��algebra C��G�Q� is the crossed product C��G�C��Q�� ��de�ned by the automorphic action ������
Conventionally� the G�action L on Q is not indicated in the notation C��G�Q�� although theconstruction clearly depends on it
One may identify L��G�C��Q�� with a subspace of the space of all �measurable� functions fromG � Q to C � an element f of the latter de�nes F � L��G�C��Q�� by F �x� � f�x� �� Clearly�L��G�C��Q�� is then identi�ed with the space of all such functions f for which
k f k��ZG
dx supq�Q
jf�x� q�j ������
is �nite� cf ����� In this realization� the operations ����� and ����� read
f � g�x� q� �
ZG
dy f�y� q�g�y��x� y��q�� ������
f��x� q� � f�x��� x��q�� ������
As always� G is here assumed to be unimodular Here is a simple example
Proposition ��� Let a locally compact group G act on Q � G by L�x� y� �� xy� ThenC��G�G� � B��L
��G�� as C��algebras�
We start from Cc�G�G�� regarded as a dense subalgebra of C��G�G� We de�ne a linear map� � Cc�G�G� � B�L��G�� by
��f���x� ��
ZG
dy f�xy��� x���y�� ���� �
One veri�es from ������ and ������ that ��f���g� � ��f � g� and ��f�� � ��f��� so that � isa representation of the ��algebra Cc�G � G� It is easily veri�ed that the Hilbert�Schmidt�norm������ of ��f� is
k ��f� k���ZG
ZG
dx dy jf�xy��� x�j�� ������
Since this is clearly �nite for f � Cc�G � G�� we conclude from ������ that ��Cc�G � G�� B��L
��G�� Since ��Cc�G � G�� is dense in B��L��G�� in the Hilbert�Schmidt�norm �which is astandard fact of Hilbert space theory�� and B��L��G�� is dense in B��L��G�� in the usual operatornorm �since by De�nition ���� even Bf �L��G�� is dense in B��L��G���� we conclude that theclosure of ��Cc�G�G�� in the operator norm coincides with B��L
��G��Since � is evidently faithful� the equality ��C��G�G�� � B��L��G��� and therefore the iso�
morphism C��G�G� � B��L��G��� follows from the previous paragraph if we can show that thenorm de�ned by ������ coincides with the operator norm of ���� This� in turn� is the case if allirreducible representations of the ��algebra Cc�G�G� are unitarily equivalent to �
To prove this� we proceed as in Proposition ���� in which we take !A � Cc�G � G�� !B � C �and !E � Cc�G� The pre�Hilbert C��module Cc�G� � C is de�ned by the obvious C �action onCc�G�� and the inner product
h���iC �� �����L��G� ������
The left�action of !A on !E is � as de�ned in ���� �� whereas the Cc�G �G��valued inner product
on !Cc�G� is given byh���iCc�G�G �� ��y���x��y�� ������
It is not necessary to consider the bounds ����� and ��� � Following the proof of Theorem� �� one shows directly that there is a bijective correspondence between the representations ofCc�G�G� and of C �
�� � HILBERT C��MODULES AND INDUCED REPRESENTATIONS
���� The abstract transitive imprimitivity theorem
We specialize to the case where Q � G�H � where H is a closed subgroup of G� and the G�actionon G�H is given by ������ This leads to the transformation group C��algebra C��G�G�H�
Theorem ���� The transformation group C��algebra C��G�G�H� is Morita�equivalent to C��H��
We need to construct a full Hilbert C��module E � C��H� for which C�� �E � C��H�� is isomor�phic to C��G�G�H� This will be done on the basis of Proposition ��� For simplicity we assumethat both G and H are unimodular In ��� we take
� !A � Cc�G�G�H�� seen as a dense subalgebra of A � C��G�G�H� as explained prior to�������
� !B � Cc�H�� seen as a dense subalgebra of B � C��H��
� !E � Cc�G�
We make a pre�Hilbert Cc�H��module Cc�G�� Cc�H� by means of the right�action
�R�f�� � �f � x�ZH
dh��xh���f�h�� ������
Here f � Cc�H� and � � Cc�G� The Cc�H��valued inner product on Cc�G� is de�ned by
h���iCc�H � h�ZG
dx��x���xh�� ������
Interestingly� both formulae may be written in terms of the right�regular representation UR of Hon L��G�� given by
UR�h���x� �� ��xh�� ������
Namely� one has
�R�f� �
ZH
dh f�h�U�h���� ������
which should be compared with ������ and
h���iCc�H � h� ��� U�h���L��G� ������
The properties ���� and ����� are easily veri�ed from �� �� and ������ respectively To prove������ we take a vector state � on C��H�� with corresponding unit vector ( � H Hence forf � Cc�H� � L��H� one has
��f� � �(� ��f�(� �
ZH
dh f�h��(� U�h�(�� ������
where U is the unitary representation of H corresponding to ��C��H��� see Theorem �����with G � H� We note that the Haar measure on G and the one on H de�ne a unique measure� on G�H � satisfying Z
G
dx f�x� �
ZG�H
d��q�
ZH
dh f�s�q�h� ���� �
for any f � Cc�G�� and any measurable map s � G�H � G for which � �s � id �where � � G� G�His the canonical projection ��x� �� �x H � xH� Combining ������� ������� and ���� �� we �nd
��h���iCc�H� �
ZG�H
d��q� kZH
dh��s�q�h�U�h�( k� � ������
Since this is positive� this proves that ��h���iCc�H� is positive for all representations � ofC��H�� so that h���iCc�H is positive in C��H� by Corollary ���� This proves ����� Condition
���� Induced group representations ��
����� easily follows from ������ as well� since h���iCc�H � � implies that the right�hand sideof ������ vanishes for all � This implies that the function �q� h� � ��s�q�h� vanishes almosteverywhere for arbitrary sections s Since one may choose s so as to be piecewise continuous� and� � Cc�G�� this implies that � � �
We now come to the left�action �L of !A � Cc�G�G�H� on Cc�G� and the Cc�G�G�H��valuedinner product h � iCc�G�G�H on Cc�G� These are given by
�L�f���x� �
ZG
dy f�xy��� �x H ���y�� ������
h���iCc�G�G�H � �x� �y H � �ZH
dh��yh���x��yh�� ������
Using ������ and ������� one may check that �L is indeed a left�action� and that Cc�G� �Cc�G�G�H� is a pre�Hilbert C��module with respect to the right�action of Cc�G�G�H� given by�R�f�� �� �L�f���� cf ��� Also� using ������� ������� ������� and ������� it is easy to verifythe crucial condition �����
To complete the proof� one needs to show that the Hilbert C��modules Cc�G� � Cc�H� andCc�G�� Cc�G�G�H� are full� and that the bounds ����� and ��� � are satis�ed This is indeedthe case� but an argument that is su�ciently elementary for inclusion in these notes does notseem to exist Enthusiastic readers may �nd the proof in MA Rie�el� Induced representations ofC��algebras� Adv� Math� �� ������ �� ���� �
���� Induced group representations
The theory of induced group representations provides a mechanism for constructing a unitaryrepresentation of a locally compact group G from a unitary representation of some closed subgroupH Theorem ���� then turns out to be equivalent to a complete characterization of inducedgroup representations� in the sense that it gives a necessary and su�cient criterion for a unitaryrepresentation to be induced
In order to explain the idea of an induced group representation from a geometric point of view�we return to Proposition �� The group G acts on the Hilbert bundle H de�ned by ���� bymeans of
U�x� � �y� v H � �xy� v H � ������
Since the left�action x � y � xy of G on itself commutes with the right�action h � y � yh of H onG� the action ������ is clearly well de�ned
The G�action U on the vector bundle H induces a natural G�action U � on the space of
continuous sections ��H� of H� de�ned on �� � ��H� by
U ��x����q� �� U�x����x��q��� ������
One should check that U ��x��� is again a section� in that ��U ��x����q�� � q� see �� �This section is evidently continuous� since the G�action on G�H is continuous
There is a natural inner product on the space of sections ��H�� given by
������� ��
ZG�H
d��q� ����q�����q��� ������
where � is the measure on G�H de�ned by ���� �� and � � � is the inner product in the �ber��� �q� � H Note that di�erent identi�cations of the �ber with H lead to the same innerproduct The Hilbert space L��H� is the completion of the space �c�H
� of continuous sectionsof H with compact support �in the norm derived from this inner product�
When the measure � is G�invariant �which is the case� for example� when G and H are uni�modular�� the operator U ��x� de�ned by ������ satis�es
�U ��x���� U ��x���� � �������� ������
�� � HILBERT C��MODULES AND INDUCED REPRESENTATIONS
When � fails to be G�invariant� it can be shown that it is still quasi�invariant in the sense that ����and ��x���� have the same null sets for all x � G Consequently� the Radon�Nikodym derivativeq � d��x���q���d��q� exists as a measurable function on G�H One then modi�es ������ to
U ��x����q� ��
sd��x���q��
d��q�U�x����x��q��� ������
Proposition ���� Let G be a locally compact group with closed subgroup H and let U be aunitary representation of H on a Hilbert space H� De�ne the Hilbert space L
��H� of L��sectionsof the Hilbert bundle H as the completion of �c�H
� in the inner product ����� where themeasure � on G�H is de�ned by ������
The map x � U ��x� given by ����� with ����� de�nes a unitary representation of G onL��H�� When � is G�invariant the expression ����� simpli�es to ������
One easily veri�es that the square�root precisely compensates for the lack of G�invariance of ��guaranteeing the property ������ Hence U ��x� is isometric on �c�H
�� so that it is bounded�and can be extended to L��H� by continuity Since U ��x� is invertible� with inverse U ��x����it is therefore a unitary operator The property U ��x�U ��y� � U ��xy� is easily checked �
The representation U ��G� is said to be induced by U�H�
Proposition ���� In the context of ������ de�ne a representation !���C��G�H�� on L��H�by
!��� !f����q� �� !f�q����q�� ������
The pair �U ��G�� !���C��G�H��� is a covariant representation of the C��dynamical system�G�C��G�H�� �� where � is given by ������
Given ����� this follows from a simple computation �
Note that the representation ������ is nothing but the right�action ���� of �C��G�H�� onL��H�� this right�action is at the same time a left�action� because �C��G�H�� is commutative
We now give a more convenient unitarily equivalent realization of this covariant representationFor this purpose we note that a section �� � Q � H
of the bundle H may alternatively berepresented as a map � � G� H which is H�equivariant in that
��xh��� � U�h���x�� ���� �
Such a map de�nes a section �� by
�����x�� � �x���x� H � ������
where � � G � G�H is given by ���� The section �� thus de�ned is independent of the choiceof x � ������x�� because of ���� �
For �� to lie in �c�H�� the projection of the support of � from G to G�H must be compact
In this realization the inner product on �c�H� is given by
����� ��
ZG�H
d����x�� ���x����x��� ������
the integrand indeed only depends on x through ��x� because of ���� �
Denition ���� The Hilbert space H is the completion in the inner product ����� of the setof continuous functions � � G � H which satisfy the equivariance condition ����� and theprojection of whose support to G�H is compact�
���� Mackey s transitive imprimitivity theorem ��
Given ������� we de�ne the induced G�action U on � by
�y� U�x���y� H �� U ��x������y��� ������
Using ������� ������� and ������� as well as the de�nition x��y� � x�y H � �xy H � ��xy� of theG�action on G�H �cf ������ we obtain
U ��x������y�� � U�x����x����y��� � U
�x��x��y���x��y� H � �y���x��y� H �
Hence we infer from ������ that
U�y���x� � ��y��x�� ������
Replacing ������ by ������ in the above derivation yields
U�y���x� �
sd����y��x��
d����x����y��x�� ������
Similarly� in the realization H the representation ������ reads
!�� !f���x� �� !f��x H ���x�� ������
Analogous to ����� we then have
Proposition ���� In the context of ������ de�ne a representation !��C��G�H�� on H cf������� by ������ The pair �U�G�� !��C��G�H��� where U is given by ����� is a covariantrepresentation of the C��dynamical system �G�C��G�H�� �� where � is given by ������
This pair is unitarily equivalent to the pair �U ��G�� !���C��G�H��� by the unitary map V �H � H� given by
V ����x�� �� �x���x� H � ������
in the sense thatV U�y�V �� � U ��y� ������
for all y � G andV !�� !f�V �� � !��� !f� ������
for all !f � C��G�H��
Comparing ������ with ������� it should be obvious from the argument leading from ������to ������ that ������ holds An analogous but simpler calculation shows ������ �
���� Mackey�s transitive imprimitivity theorem
In the preceding section we have seen that the unitary representation U�G� induced by a unitaryrepresentation U of a closed subgroup H � G can be extended to a covariant representation�U�G�� !��C��G�H�� The original imprimitivity theorem of Mackey� which historically precededTheorems � � and ����� states that all covariant pairs �U�G�� !��C��G�H�� arise in this way
Theorem ���� Let G be a locally compact group with closed subgroup H and consider theC��dynamical system �G�C��G�H�� �� where � is given by ������ Recall cf� ����� that a co�variant representation of this system consists of a unitary representation U�G� and a representation!��C��G�H�� satisfying the covariance condition
U�x�!�� !f�U�x��� � !�� !fx� ���� �
for all x � G and !f � C��G�H� here !fx�q� �� !f�x��q��Any unitary representation U�H� leads to a covariant representation �U�G�� !��C��G�H�� of
�G�C��G�H�� �� given by ������ ����� and ������ Conversely any covariant representation�U� !�� of �G�C��G�H�� �� is unitarily equivalent to a pair of this form�
This leads to a bijective correspondence between the space of equivalence classes of unitaryrepresentations of H and the space of equivalence classes of covariant representations �U� !�� ofthe C��dynamical system �G�C��G�H�� �� which preserves direct sums and therefore irreducibilityhere the equivalence relation is unitary equivalence�
�� � HILBERT C��MODULES AND INDUCED REPRESENTATIONS
The existence of the bijective correspondence with the stated properties follows by combiningTheorems ���� and � �� which relate the representations of C��H� and C��G�G�H�� with The�orems ���� and ���� which allow one to pass from ��C��H�� to U�H� and from ��C��G�G�H��to �U�G�� !��C��G�H��� respectively
The explicit form of the correspondence remains to be established Let us start with a technicalpoint concerning Rie�el induction in general Using ������ ������ and ������ one shows thatk !V � k�k � k� where the norm on the left�hand side is in !H� and the norm on the right�handside is the one de�ned in ����� It follows that the induced space !H obtained by Rie�el�inducingfrom a pre�Hilbert C��module is the same as the induced space constructed from its completionThe same comment� of course� applies to H
We will use a gerenal technique that is often useful in problems involving Rie�el induction
Lemma ���� Suppose one has a Hilbert space H� with inner product denoted by � � �� and a
linear map !U � E �H � H� satisfying
� !U !�� !U !��� � � !�� !��� ������
for all !�� !� � E �H�
Then !U quotients to an isometric map between E �H�N and the image of !U in H� � When
the image is dense this map extends to a unitary isomorphism U � H � H� � Otherwise U is
unitary between H and the closure of the image of !U �In any case the representation ��C��E �B�� is equivalent to the representation �� �C��E �B��
de�ned by continuous extension of
�� �A� !U !� �� !U�A� I !��� ������
It is obvious that N � ker� !U�� so that� comparing with ������ one indeed has U � � ��� � U �
We use this lemma in the following way To avoid notational confusion� we continue to de�note the Hilbert space H de�ned in Construction ���� starting from the pre�Hilbert C��moduleCc�G�� Cc�H� de�ned in the proof of ����� by H The Hilbert space H de�ned below �������however� will play the role H
� in ����� and will therefore be denoted by this symbolConsider the map !U � Cc�G� �H � H
� de�ned by linear extension of
!U�� v�x� ��
ZH
dh��xh�U�h�v� ������
Note that the equivariance condition ���� � is indeed satis�ed by the left�hand side� as followsfrom the invariance of the Haar measure
Using ������ ������� and ������ with G� H � one obtains
��� v��� w�� �
ZH
dh ��� UR�h���L��G�v� U�h�w� �
ZH
dh
ZG
dx��x���xh��v� U�h�w��
��� ��cf ������ On the other hand� from ������ and ������ one has
� !U�� v� !U�� w�H��
�
ZH
dh
ZG�H
d����x��
ZH
dk��xk���xh��U�k�v� U�h�w�� ��� ��
Shifting h � kh� using the invariance of the Haar measure on H � and using ���� �� one veri�es������ It is clear that !U�Cc�G� �H� is dense in H� so by Proposition ���� one obtains thedesired unitary map U � H � H
� Using ������ and ������� one �nds that the induced representation of C��G�G�H� on H
� isgiven by
��f���x� �
ZG
dy f�xy��� �x H ���y�� ��� ��
��
this looks just like ������� with the di�erence that � in ������ lies in Cc�G�� whereas � in��� �� lies in H
� Indeed� one should check that the function ��f�� de�ned by ��� �� satis�esthe equivariance condition ���� �
Finally� it is a simple exercise the verify that the representation ��C��G�G�H�� de�ned by��� �� corresponds to the covariant representation �U�G�� !��C��G�H�� by the correspondence������ � ���� � of Theorem ��� �
� Applications to quantum mechanics
��� The mathematical structure of classical and quantum mechanics
In classical mechanics one starts from a phase space S� whose points are interpreted as the purestates of the system More generally� mixed states are identi�ed with probability measures on SThe observables of the theory are functions on S� one could consider smooth� continuous� bounded�measurable� or some other other class of real�vaued functions Hence the space AR of observablesmay be taken to be C��S�R�� C��S�R�� Cb�S�R�� or L��S�R�� etc
There is a pairing h � i � S �AR� R �� between the state space S of probability measures �on S and the space AR of observables f This pairing is given by
h�� fi �� ��f� �
ZS
d�� � f� �� ����
The physical interpretation of this pairing is that in a state � the observable f has expectationvalue h�� fi In general� this expectation value will be unsharp� in that h�� fi� � h�� f�i However�in a pure state �seen as the Dirac measure �� on S� the observable f has sharp expectation value
���f� � f� �� ����
In elementary quantum mechanics the state space consists of all density matrices � on someHilbert space H� the pure states are identi�ed with unit vectors � The observables are taken tobe either all unbounded self�adjoint operators A on H� or all bounded self�adjoint operators� or allcompact self�adjoint operators� etc This time the pairing between states and observables is givenby
h��Ai � Tr �A� ����
In a pure state � one hash�� Ai � ��� A��� ����
A key di�erence between classical and quantum mechanics is that even in pure states expectationvalues are generally unsharp The only exception is when an observable A has discrete spectrum�and � is an eigenvector of A
In these examples� the state space has a convex structure� whereas the set of observables is areal vector space �barring problems with the addition of unbounded operators on a Hilbert space�We may� therefore� say that a physical theory consists of
� a convex set S� interpreted as the state space�
� a real vector space AR� consisting of the observables�
� a pairing h � i � S �AR� R ��� which assigns the expectation value h�� fi to a state � andan observable f
In addition� one should specify the dynamics of the theory� but this is not our concern hereThe situation is quite neat if S and AR stand in some duality relation For example� in the
classical case� if S is a locally compact Hausdor� space� and we take AR � C��S�R�� then thespace of all probability measures on S is precisely the state space of A � C��S� in the sense ofDe�nition ���� see Theorem ��� In the same sense� in quantum mechanics the space of alldensity matrices on H is the state space of the C��algebra B��H� of all compact operators on H�
�� � APPLICATIONS TO QUANTUM MECHANICS
see Corollary ������ On the other hand� with the same choice of the state space� if we take ARto be the space B�H�R of all bounded self�adjoint operators on H� then the space of observablesis the dual of the �linear space spanned by the� state space� rather then vice versa� see Theorem����
In the C��algebraic approach to quantum mechanics� a general quantum system is speci�edby some C��algebra A� whose self�adjoint elements in AR correspond to the observables of thetheory The state space of AR is then given by De�nition ��� This general setting allows for theexistence of superselection rules We will not go into this generalization of elementary quantummechanics here� and concentrate on the choice A � B�H�
��� Quantization
The physical interpretation of quantum mechanics is a delicate matter Ideally� one needs to specifythe physical meaning of any observable A � AR In practice� a given quantum system arises from aclassical system by �quantization� This means that one has a classical phase space S and a linearmap Q � A�
R� L�H�� where A�
Rstands for C��S�R�� or C��S�R�� etc� and L�H� denotes some
space of self�adjoint operators on H� such as B��H�R or B�H�R Given the physical meaning ofa classical observable f � one then ascribes the same physical interpretation to the correspondingquantum observable Q�f� This provides the physical meaning of al least all operators in theimage of Q It is desirable �though not strictly necessary� that Q preserves positivity� as well asthe �approximate� unit
It is quite convenient to assume that A�R
� C��S�R�� which choice discards what happens atin�nity on S We are thus led to the following
Denition ��� Let X be a locally compact Hausdor� space� A quantization of X consists ofa Hilbert space H and a positive map Q � C��X� � B�H�� When X is compact it is required thatQ��X� � I and when X is non�compact one demands that Q can be extended to the unitizationC��X�I by a unit�preserving positive map�
Here C��X� and B�H� are� of course� regarded as C��algebras� with the intrinsic notion ofpositivity given by � � Also recall De�nition ��� of a positive map It follows from � � thata positive map automatically preserves self�adjointness� in that
Q�f� � Q�f�� ����
for all f � C��X�� this implies that f � C��X�R� is mapped into a self�adjoint operatorThere is an interesting reformulation of the notion of a quantization in the above sense
Denition ��� Let X be a set with a �algebra * of subsets of X� A positive�operator�valued measure or POVM on X in a Hilbert space H is a map $ � A�$� from * to B�H��
the set of positive operators on H satisfying A��� � � A�X� � I and A��i$i� �P
i A�$i� forany countable collection of disjoint $i � * where the in�nite sum is taken in the weak operatortopology�
A projection�valued measure or PVM is a POVM which in addition satis�es A�$��$�� �A�$��A�$�� for all $��$� � *�
Note that the above conditions force � � A�$� � I A PVM is usually written as $ � E�$��it follows that each E�$� is a projection �take $� � $� in the de�nition� This notion is familiarfrom the spectral theorem
Proposition ��� Let X be a locally compact Hausdor� space with Borel structure *� There isa bijective correspondence between quantizations Q � C��X� � B�H� and POVM�s $ � A�$� onS in H given by
Q�f� �
ZS
dA�x� f�x�� �� �
The map Q is a representation of C��X� i� $ � A�$� is a PVM�
��� Stinespring s theorem and coherent states ��
The precise meaning of �� � will emerge shortly Given the assumptions� in view of ��� and�� we may as well assume that X is compact
Given Q� for arbitrary � � H one constructs a functional ����� on C�X� by ������f� �����Q�f��� Since Q is linear and positive� this functional has the same properties Hence theRiesz representation theorem yields a probability measure ���� on X For $ � * one then puts��� A�$��� �� �����$�� de�ning an operator A�$� by polarization The ensuing map $ � A�$�is easily checked to have the properties required of a POVM
Conversely� for each pair ��� � H a POVM $ � A�$� in H de�nes a signed measure ��� on X by means of ��� �$� �� ��� A�$��� This yields a positive map Q � C�X� � B�H� by���Q�f��� ��
RX d��� �x� f�x�� the meaning of �� � is expressed by this equation
Approximating f� g � C�X� by step functions� one veri�es that the property E�$�� � E�$� isequivalent to Q�fg� � Q�f�Q�g� �
Corollary ��� Let $ � A�$� be a POVM on a locally compact Hausdor� space X in a Hilbertspace H� There exist a Hilbert space H a projection p on H a unitary map U � H � pH and a PVM $ � E�$� on H such that UA�$�U�� � pE�$�p for all $ � *�
Combine Theorem ���� with Proposition ��� �
When X is the phase space S of a physical system� the physical interpretation of the map$ � A�$� is contained in the statement that the number
p��$� �� Tr �A�$� ����
is the probability that� in a state �� the system in question is localized in $ � SWhen X is a con�guration space Q� it is usually su�cient to take the positive map Q to be
a representation � of C��Q� on H By Proposition ���� the situation is therefore described by aPVM $ � E�$� on Q in H The probability that� in a state �� the system is localized in $ � Qis
p��$� �� Tr �E�$�� ����
��� Stinespring�s theorem and coherent states
By Proposition ����� a quantization Q � C��X� � B�H� is a completely positive map� andDe�nition ��� implies that the conditions for Stinespring�s Theorem ���� are satis�ed Wewill now construct a class of examples of quantization in which one can construct an illuminatingexplicit realization of the Hilbert space H and the partial isometry W
Let S be a locally compact Hausdor� space �interpreted as a classical phase space�� and consideran embedding � �� of S into some Hilbert space H� such that each �� has unit norm �so thata pure classical state is mapped into a pure quantum state� Moreover� there should be a measure� on S such that Z
S
d�� ����������� ���� � �������� ����
for all ����� � H The �� are called coherent states for SCondition ���� guarantees that we may de�ne a POVM on S in H by
A�$� �
Z�
d�� � ��� � �����
where �� is the projection onto the one�dimensional subspace spanned by � �in Dirac�s notationone would have �� � j� � �j�
The positive map Q corresponding to the POVM $ � A�$� by Proposition ��� is given by
Q�f� �
ZS
d�� � f� ���� � �����
In particular� one has Q��S� � I
� � APPLICATIONS TO QUANTUM MECHANICS
For example� when S � T �R� � R� � so that � �p� q�� one may take
��p�q�x� � ����n��e��
�ipq�ipxe��x�q
��� �����
in H � L��R� � Eq ���� then holds with d��p� q� � d�pd�q������ Extending the map Q fromC��S� to C��S� in a heuristic way� one �nds that Q�qi� and Q�pi� are just the usual position� andmomentum operators in the Schr�odinger representation
In Theorem ���� we now put A � C��S�� B � B�H�� ��A� � A for all A We may thenverify the statement of the theorem by taking
H � L��S� d��� �����
The map W � H � H is then given by
W�� � �� ������� �����
It follows from ���� that W is a partial isometry The representation ��C��S�� is given by
��f��� � � f� ��� �� �����
Finally� for ������ one has the simple expression
!Q�f� � UQ�f�U�� � pfp� ��� �
Eqs ����� and ��� � form the core of the realization of quantum mechanics on phase spaceOne realizes the state space as a closed subspace of L��S� �de�ned with respect to a suitablemeasure�� and de�nes the quantization of a classical observable f � C��S� as multiplication by f �sandwiched between the projection onto the subspace in question This should be contrasted withthe usual way of doing quantum mechanics on L��Q�� where Q is the con�guration space of thesystem
In speci�c cases the projection p � WW � can be explicitly given as well For example� in thecase S � T �R� considered above one may pass to complex variables by putting z � �q � ip��
p�
We then map L��T �R� � d�pd�q������� into K �� L��C � � d�zd�z exp��zz�����i��� by the unitaryoperator V � given by
V ��z� z� �� e�
�zz��p � �z � z��
p�� q � �z " z��
p��� �����
One may then verify from ����� and ����� that V pV �� is the projection onto the space of entirefunctions in K
��� Covariant localization in con�guration space
In elementary quantum mechanics a particle moving on R� with spin j � N is described by theHilbert space
HjQM
� L��R� ��Hj � �����
where Hj � C �j�� carries the irreducible representation Uj�SO���� �usually called Dj� The basicphysical observables are represented by unbounded operators QS
k �position�� PSk �momentum�� and
JSk �angular momentum�� where k � �� �� � These operators satisfy the commutation relations�say� on the domain S�R� ��Hj�
�QSk � Q
Sl � �� �����
�PSk � Q
Sl � �i��kl� �����
�JSk � QSl � i��klmQ
Sm� �����
�PSk � P
Sl � �� �����
�JSk � JSl � i��klmJ
Sm� �����
�JSk � PSl � i��klmP
Sm� �����
��� Covariant localization in con�guration space ��
justifying their physical interpretationThe momentum and angular momentum operators are most conveniently de�ned in terms of a
unitary representation U jQM of the Euclidean group E��� � SO���n R� on Hj
QM� given by
U jQM
�R� a���q� � Uj�R���R���q � a��� �����
In terms of the standard generators Pk and Tk of R� and SO���� respectively� one then has PSk �
i�dU jQM�Pk� and JSk � i�dU j
QM�Tk�� see ������ The commutation relations ����� � ����� followfrom ������ and the commutation relations in the Lie algebra of E���
Moreover� we de�ne a representation !�jQM of C��R� � on Hj
QM by
!�jQM
� !f� � !f � Ij� ��� �
where !f is seen as a multiplication operator on L��R� � The associated PVM $ � E�$� on R�
in HjQM �see ���� is E�$� � �� � Ij� in terms of which the position operators are given by
QSk �RR�dE�x�xk � cf the spectral theorem for unbounded operators Eq ����� then re#ects the
commutativity of C��R� �� as well as the fact that !�jQM is a representationIdentifying Q � R� with G�H � E����SO��� in the obvious way� one checks that the canonical
left�action of E��� on E����SO��� is identi�ed with its de�ning action on R� It is then not hard toverify from ����� that the pair �U j
QM�E����� !�jQM�C��R� ��� is a covariant representation of the C��dynamical system �E���� C��R� �� ��� with � given by ������ The commutation relations ����������� are a consequence of the covariance relation ���� �
Rather than using the unbounded operators QSk � PS
k � and JSk � and their commutation rela�
tions� we therefore state the situation in terms of the pair �U jQM�E����� !�jQM�C��R� ��� Such a
pair� or� equivalently� a non�degenerate representation �jQM of the transformation group C��algebraC��E����R� � �cf ���� then by de�nition describes a quantum system which is localizable in R� and covariant under the de�ning action of E��� It is natural to require that �jQM be irreducible�in which case the quantum system itself is said to be irreducible
Proposition ��� An irreducible quantum system which is localizable in R� and covariant underE��� is completely characterized by its spin j � N� The corresponding covariant representation�U j�E����� !�j�C��R
� ��� given by ������ ����� and ����� is equivalent to the one describedby ���� ���� and �����
This follows from Theorem ���� The representation U jQM�E���� de�ned in ����� is unitarily
equivalent to the induced representation U j To see this� check that the unitary map V � Hj �Hj
QM de�ned by V �j�q� �� �j�e� q� intertwines U j and U jQM In addition� it intertwines the
representation ������ with !�jQM as de�ned in ��� � �
This is a neat explanation of spin in quantum mechanicsGeneralizing this approach to an arbitrary homogeneous con�guration space Q � G�H � a non�
degenerate representation � of C��G�G�H� on a Hilbert space H describes a quantum systemwhich is localizable in G�H and covariant under the canonical action of G on G�H By ��� thisis equivalent to a covariant representation �U�G�� !��C��G�H��� on H� and by Proposition ���one may instead assume one has a PVM $ � E�$� on G�H in H and a unitary representationU�G�� which satisfy
U�x�E�$�U�x��� � E�x$� �����
for all x � G and $ � *� cf ����� The physical interpretation of the PVM is given by �����the operators de�ned in ������ play the role of quantized momentum observables GeneralizingProposition ���� we have
Theorem ��� An irreducible quantum system which is localizable in Q � G�H and covariantunder the canonical action of G is characterized by an irreducible unitary representation of H� Thesystem of imprimitivity �U�G�� !�j �C��G�H��� is equivalent to the one described by ����� and������
�� � APPLICATIONS TO QUANTUM MECHANICS
This is immediate from Theorem ���� �
For example� writing the two�sphere S� as SO����SO���� one infers that SO����covariant quan�tum particles on S� are characterized by an integer n � Z For each unitary irreducible represen�tation U of SO��� is labeled by such an n� and given by Un��� � exp�in��
��� Covariant quantization on phase space
Let us return to quantization theory� and ask what happens in the presence of a symmetry groupThe following notion� which generalizes De�nition ���� is natural in this context
Denition ��� A generalized covariant representation of a C��dynamical system�G�C��X�� �� where � arises from a continuous G�action on X by means of ����� consists of apair �U�Q� where U is a unitary representation of G on a Hilbert space H and Q � C��X� � B�H�is a quantization of C��X� in the sense of De�nition ����� which for all x � G and !f � C��X�satis�es the covariance condition
U�x�Q� !f�U�x�� � Q��x� !f��� �����
This condition may be equivalently stated in terms of the POVM $ � A�$� associated to Q�cf ���� by
U�x�A�$�U�x��� � A�x$�� �����
Every �ordinary� covariant representation is evidently a generalized one as well� since a rep�resentation is a particular example of a quantization A class of examples of truly generalizedcovariant representations arises as follows Let �U�G�� !��C��G�H�� be a covariant representationon a Hilbert space K� and suppose that U�G� is reducible Pick a projection p in the commu�tant of U�G�� then �pU�G�� p!�p� is a generalized covariant representation on H � pK Of course��U� !�� is described by Theorem ����� and must be of the form �U� !�� This class actually turnsout to exhaust all possibilities What follows generalizes Theorem ���� to the case where therepresentation !� is replaced by a quantization Q
Theorem ��� Let �U�G��Q�C��G�H��� be a generalized covariant representation of the C��dynamical system �G�C��G�H�� �� de�ned with respect to the canonical G�action on G�H�
There exists a unitary representation U�H� with corresponding covariant representation �U� !��of �G�C��G�H�� �� on the Hilbert space H as described by ������ ����� and ����� and a pro�jection p on H in the commutant of U�G� such that �pU�G�� p!�p� and �U�G��Q�C��G�H���are equivalent�
We apply Theorem ���� To avoid confusion� we denote the Hilbert space H and the repre�sentation � in Construction ���� by !H and !�� respectively� the space de�ned in ���� and theinduced representation ������ will still be called H and �� as in the formulation of the theoremabove Indeed� our goal is to show that �!�� !H� may be identi�ed with ���H� We identify Bin ���� and ���� with B�H�� where H is speci�ed in ���� we therefore omit the representation� occurring in ���� etc� putting H � H
For x � G we de�ne a linear map !U�x� on C��G�H��H by linear extension of
!U�x�f �� �� �x�f�� U�x��� �����
Since �x � �y � �xy� and U is a representation� !U is clearly a G�action Using the covariancecondition ����� and the unitarity of U�x�� one veri�es that
� !U�x�f ��� !U�x�g ���� � �f ��� g � ��� � �����
where � � �� is de�ned in ������ Hence !U�G� quotients to a representation !U�G� on !H Com�puting on C��G�H��H and then passing to the quotient� one checks that � !U� !�� is a covariantrepresentation on !H By Theorem ����� this system must be of the form �U� !�� �up to unitaryequivalence�
Finally� the projection p de�ned in ���� commutes with all !U�x� This is veri�ed from������� ������� and ����� The claim follows �
LITERATURE ��
Literature
Bratteli� O and DW Robinson ����� Operator Algebras and Quantum Statistical Mechanics Vol�I� C�� and W ��Algebras Symmetry Groups Decomposition of States� �nd ed Springer� Berlin
Bratteli� O and DW Robinson ����� Operator Algebras and Quantum Statistical Mechanics Vol�II� Equilibrium States Models in Statistical Mechanics Springer� Berlin
Connes� A ����� Noncommutative Geometry Academic Press� San Diego
Davidson� KR ���� C��Algebras by Example Fields Institute Monographs � Amer Math Soc�Providence �RI�
Dixmier� J ����� C��Algebras North�Holland� Amsterdam
Fell� JMG ����� Induced Representations and Banach ��algebra Bundles� Lecture Notes in Math�ematics ��� Springer� Berlin
Fell� JMG and RS Doran ����� Representations of ��Algebras Locally Compact Groups andBanach ��Algebraic Bundles Vol� � Academic Press� Boston
Kadison� RV ����� Operator algebras � the �rst forty years In� Kadison� RV �ed� OperatorAlgebras and Applications� Proc� Symp� Pure Math� ������ pp ���� American MathematicalSociety� Providence
Kadison� RV ����� Notes on the Gelfand�Neumark theorem In� Doran� RS �ed� C��algebras���������� Cont� Math� ���� pp ����� Amer Math Soc� Providence �RI�
Kadison� RV and JR Ringrose ����� Fundamentals of the Theory of Operator Algebras I Aca�demic Press� New York
Kadison� RV and JR Ringrose ���� Fundamentals of the Theory of Operator Algebras IIAcademic Press� New York
Lance� EC ����� Hilbert C��Modules� A Toolkit for Operator Algebraists� LMS Lecture Notes��� Cambridge University Press� Cambridge
Mackey� GW ��� � The Mathematical Foundations of Quantum Mechanics New York� Benjamin
Mackey� GW ��� � Induced Representations Benjamin� New York
Mackey� GW ����� Unitary Group Representations in Physics Probability and Number TheoryBenjamin� New York
Pedersen� GK ����� C��Algebras and their Automorphism Groups Academic Press� London
Pedersen� GK ����� Analysis Now Springer� New York
Takesaki� M ����� Theory of Operator Algebras I Springer� Heidelberg
Wegge�Olsen� NE ����� K�theory and C��algebras Oxford University Press� Oxford