of
this publication may be reproduced. stored in a retrieval
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recording or otherwise, without the prior permission of the copyright owner,
North-Holland ISBN for the series: 0 7204 2450 X
North-Holland ISBN for
A translation of:
' BORDAS (Gauthier-Villars), Paris. 1969
Published by:
ELSEVIER NORTH-HOLLAND. INC.
52 Vanderbilt Avenue
Dixmier, Jacques.
C*-algebras
Bibliography: p. 410
3. Representations of groups. I. Title.
QA326.D5213 512’.55 77-7133 ISBN 0-7204-0762-1
PRINTED IN THE NETHERLANDS
INTRODUCTION
Let H be a Hilbert space and :£(H) the set of bounded linear
operators on H. Let A be a subset of :£(H) which is closed under
addition, multiplication, multiplication by scalars and under the adjoint
operation, and which is also a closed subset of :£(H) in the operator›
norm topology. A is then a special type of involutive Banach algebra;
such an algebra is called a C*-algebra.
The
theory of C*-algebras had its origins in 1943 after
Gelfand and
Naimark showed that among all involutive Banach algebras, C*-algebras
could be characterized by a few simple axioms.
I t
was subsequently
seen C*-algebras play a basic role in study of the represen›
tations of a very extensive class of involutive Banach algebras; for each
algebra B of this class, a C*-algebra A can be constructed
such
that
the
representationsof B in a Hilbert space can be identified with those of A.
For many questions (notably those involving ideals), A is easier to
handle than
when B is
the algebra of integrable functions on a locally compact group G, so that
the study of the unitary representations of G reduces to
that
of the
representations of a certain C*-algebra, called the C*-alge ra of G.
The
study of C*-algebras takes up almost four fifths of this book,
where the main results due, a ong others, to Fell, Glimm, Kadison,
Kaplansky, Mackey and Segal are expounded. I t seemed to me a pity
not to make use of the material thus accumulated together with that
contained in my book on von Neumann algebras (Cahiers Scientifiques,
fasc. XXV; referred to as [A v N]) to say a little about unitary represen›
tations of groups. All the more so as the theory of groups provides some
of the most interesting examples of C*-algebras. However, the latter
pages of the book do
not
representations. To get an idea of the questions which would
have
the
"Infinite dimensional group representations" (Bull. Amer. Math. Soc.,
 
xii
INTRODUCTION
article are concerned with questions treated in he present work. Fur›
thermore, Gelfand and the Russian School on (he one hand, and Harish›
Chandra on the other are hardly mentioned here; in other words my
exposition is very incomplete as far as groups are concerned.
Although v n Neumann algebras are examples of C*-algebras, the
theory of C*-algebras in fact depends on that of von Neumann algebras,
as will be seen time and again. To save the reader having to refer
constantly to [A v N], the essential results concerning von Neumann
algebras are assembled in Appendix A.
Furthermore, a mixed bag of results, used in the book, is grouped in
Appendix B, and the results presented here follow no particular plan.
It seemed to me, while writing [A v N], that most of the theorems
were already in a more or less definitive form. By contrast,the theory of
C*-algebras appeared to me to have a long way to go before a stable
state is reached.
Each chapter finishes with some additional results, given either
without proof or with just a sketch of the proof. Some of the easier of
these results can serve as exercises, although it would have been difficult
to distinguish these learly from the others. I have merely indicated,
with an asterisk, those resu ts whose proof is really lengthy (e.g. more
than three pages), although one should not attach too much importance
to this classification. The additional results are not used in the sub›
sequent chapters.
The bibliography includes the essential references for the theory of
C*-algebras, while being far from complete in the area of groups.
The reader of this book, as with [A v N] is assumed to be well
acquainted with general topology, topological vector spaces and in›
tegration theory. The proofs of results in C*-algebras are presented in
reasonable detail, while those dealing with groups are more condensed
and assume hat the rea er is familiar with the properties of the
convolution product. On the other hand, the reader is assumed to have a
knowledge of the theory of locally compact commutative groups; for,
while no call is, in principle, made on this theory, apart from one or two
points of detail, reference to the commutative case often sheds light on
the problems studied.
The results are numbered lexicographically. A reference such as 4.7.5
s self-explanatory. References of the form A 8, B 15 refer to Appendices
A and B.
 
INTRODUCTION
xiii
those of Naimark and Rickart. The books of Naimark and Rickart
include, among other things, the general theory of normed algebras,
which is not to be found here. On the other hand, in the special case of
C*-algebras, the present book is more detailed. Ther are, of course,
some themes common to all three books, but they are relatively few.
The small number of general results on normed algebras which are used
here are quoted in Appendix B.
Concerning the English edition, a few corrections have been made.
The bibliography has been completed up to 1975. I thank S. Berberianand
J. Brohan for their help in this connection.
 
NORMED INVOLUTIVE ALGEBRAS
1.1. Involutive algebras
1.1.1. Let A be an algebra over the field C of complex numbers. An
involution in A is a map
x
that
E C. An algebra over C endowed with an
involution is called an involutive algebra.
x* is often called the adjoint of
x. A subset of A which is closed under
the involution operation is said to
be
self-adjoint.
Property (i) implies that an involution in A is necessarily
a bijection of A onto itself.
1.1.2.
Examples
(1)
On A = C, the map z ~ z (where z is the complex conjugate of z)
is an involution with which A becomes a commutative involutive alge›
bra.
and
A
the map f
When
(3)
Let H
be a Hilbert space and A = 2(H) the algebra of continuous
endomorphisms of H. Furnished with the usual adjoint operation, A is
an involutive algebra. Examples (2) and (3) will playa fundamental role.
(Throughout the book, "Hilbert space" means "complex Hilbert
space".)
(4) Let G be a unimodular locally compact group and A the con›
volution algebra L l(G). For
each f E L l(G)
put
r.
 
1.1.3. We now introduce some terminology suggested by example (3)
above. Let A be an involutive algebra. An element x E A is said to be
hermitian if x* =
x and normal if xx’"
= x*x. An idempotent hermitian
element is called a projection. Each hermitian element is normal and the
set of hermitian elements is a real vector subspace of A.
I f x and yare
hermitian, we have (xy)* = y*x* =yx, and so xy is hermitian if x and y
commute. For every x E A, xx* and x*x are hermitian, although a
general hermitian element cannot be so represented as example (1)
1.1.4. Each x E A can be written uniquely in the form XI +
iX
2
with XI’ X2
hermitian. (In example (1) above, this expression is just the decom›
position of a complex number into its real and imaginary parts.) In fact,
if we put
2
2i
then XI and X2 are hermitian and we have x =XI +iX2’
Conversely, if
=XI +iX2 with XI and X2 hermitian, we have x* =
x
l
- ix
2
which proves our assertion. Note that
so that x is normal if and only if XI and X2 commute.
1.1.5. I f A possesses a left identity 1, we have, for each
x
x,
*
an invertible element of
=(x-Ix)* =1*=1,
so that x* is invertible, and (X*fl = (x")"; conversely, if x* is invertible,
x** =x is also invertible. Since (x - A 1)*
=x* - A. 1 for each A E C, it
follows that
SpAx* =SPAx.
 
ALGEBRAS
5
SP&X or simply Sp x, is the set of scalars
A
such
invertible.)
An element x E A is said to be unitary if xx* =x*x =1,
or
(1)
above, the unitar
elements are the complex numbers of absolute value 1.] The unitary
elements of A constitute a group under multiplication, the unitary group
of A; in fact, if x and
yare
(xy)*-l
l
1. 1.6. Let
A be an involutive algebra. I f A is the algebra obtained from
A by adjoining an identity to
,
to see that the involution on A
can be extended to A in a unique way: we define
(A, x)*
we have
Sp~x* =
Sp~x
Sp’x
where
B
is the algebraobtained from B by the adjunction of an identity.
Clearly, 0
E SPBx
for
each
x
E
B.)
1.1.7. Let A and B be two involutive algebras. A morphism (resp.
isomorphism) of A into B is a map (resp. a bijection)
q>
that q>(x +y) =q>(x) +q>(y), q>(Ax) =
Aq>(x),
q>(x)* for any x, yEA ,
A E
is a morphism of the
underlying algebra of A into the underlying algebra of B. In cases of
possible confusion, we say more precisely "morphism
for
the involutive
algebra structure" or "morphism for the algebra structure" as the case
may be.
1.1 .8. Let A be a involutive algebra. An involutive subalgebraof A is a
subalgebra of A which is closed under the involution. The intersection
of any family of involutive subalgebrasis again an involutive subalgebra;
thus, if M is any subset of A, there is a smallest involutive subalgebraof
A containing M, namely the intersection of all the involutive subalgebras
of A which contain M, and this is called the involutive subalgebra of A
generated by M; it is the set of linear comb nations of elements of the
form XIX2’
single element x, this subalgebra is commutative if and only
if X is
B
a self-adjoint two-sided ideal of
A. The involution on A induces an involution on the quotient algebra
AIB, and the canonical map of A onto AlB is a morphism.
The product of any family of involutive algebras is itself an involutive
algebra in a natural way.
The reversed algebra of an involutive algebra is itself an involutive
algebra when endowed with the same involution.
1.1.9. Let A be an involutive algebra and M a self-adjoint subset of A.
Then the commutant M’ of M in A is an involutive subalgebraof A, and
the bicommutant
A
containing
M;
A
A
elements of M commute pairwise, then
Me M’, so that M’:>Mil
and
Mil is commutative. I f x E A and M ={x, x*}, then Mil is commutative if
and only if x is normal.
1.1.10. Let A be an involutive algebra. I f I is a linear form on A, the
function x ....I(x*) on A is also a linear form on A, denoted by f* and
called the adjoint of
if
A
E
C, and I is said to be hermitian if 1=f*. Every linear form I on A
has a nique expression of the form II +i/
2
I
of hermitian elements of A. The map I .... I I
A
h
is an
isomorphism of the real vector space of hermitian forms onto the dual
space of the real vector space A
h
characterof A, then X*
References: [1101], [1323].
together with an involution
x
E A. If,
 
7
1.2.2. Examples. The four examples of 1.1.2 are examples of involutive
Banach algebras if the norms are defined as follows: in example (1)
set
[z] = [z] for each z E C; in example (2) set IIIII= SUPtEX I/(t)1 for each
I E A;
,;t(H);
set 11/11 =fa I/(g)ldg for each IE L1(G).
1.2.3. Let A be a normed involutive algebra, and A the involutive
algebra obtained from A by the adjunction of an identity. The norm on
A can be extended to A in such a way as to make A a normed involutive
algebra (for example, one can put II(A, x)11 =IA 1+ [x] for AE
C, x E A).
Any normal involutive algebra so obtained is called a normed involutive
algebra obtained from A by the adjunction of an identity.
1.2.4. Let A and B be two normed involutive algebras. A morphism of
A into B will simply mean a morphism of the u derlying involutive
algebras, without any condition on the norms. On the other hand, an
isomorphism will mean a norm-preserving isomorphism of the underly›
ing involutive algebras.
1.2.5. The closure of an involutive subalgebra of a normed involutive
algebra A is itself an involutive subalgebra. I f MeA , the smallest
closed involutive subalgebra B
containing M is called the closed involu›
tive subalgebra generated by M and is the closure of the involutive
subalgebra generated by M. I f M consists of a single normal element,
then B is commutative.
The quotient of a normed involutive algebra by a closed self-adjoint
two-sided ideal, the product of a finite number of normed involutive
algebras, the reversed algebra of a normed involutive algebra and the
completion of a normed involutive algebra are all normed involutive
algebras in a natural way.
1.2.6. Let A be a normed involutive algebra. I f
f is a continuous linear
form on A, then 1* is also continuous and 111*11 =Ilfll
as the unit ball of A
is self-adjoint. The set A
h
of hermitian elements of A is a real normed
vector space. Now let I be a continuous hermitian linear form on A and
let g=I IA
h
Then
11/11 = IIgll; in act, it is clear that Ilfll~ JlglI; on the other
hand, for each E > 0 there exists an x E A such that Ilxll,;;;;1 and I/(x)1 ~
11/11- E and multiplying x by a scalarof absolute value 1 if necessary, we
can assume that I(x) ~ O. Then
Ig(!<x+x*»1 =~1/(x) +l(x*)1=I(x)
~ 11/11- E
and since IH(x + x*)11 ,;;;; I, we see that Ilgll ~
11/11- E and the assertion
follows. The continuous hermitian linear forms on A may thus be
identified with the continuous real linear forms on A
h
1.3. C*-algebras
1.3.1. DEFINITION. A C*-algebra is an involutive Banach algebra A
such that IIxl1
=Ilx*xll for every x E A.
1.3.2. Examples (1), (2) and (3) of 1.1.2, 1.2.2 are examples of C*›
algebras. Example (4), however, is not in general an example of a
C*-algebra.
1.3.3. I f A is a C*-algebra, so is every closed involutive subalgebra of
A. In particular, if H is Hilbert space, every closed involu ive
subalgebra of 2 (H) is a C*-algebra; it will later be seen (2.6.1) that
every C*-algebrais isomorphic to a C*-algebra of this type, and it is this
example that has given rise to the theory of C*-algebras.
Let (A;)iEI be a family of C*-algebras. Let A be the set of (X;)iEI such
that Xi E Ai for each i E I and such that SUPiEI
Ilxill <+00. With the
(x;)* =
we immediately see that A is a C*-algebra, called the product C*›
algebra of the
A;’s. I t should be noted that the set
A
is not the product
(set) of the A/s.
Let A be a C*-algebra, and consider the algebra obtained from A by
replacing the multiplication
by the multiplication
(x, y) ~ yx,
while all other algebraic operations and the norm are the same as those
of A. Consider, in other words, the reversed normed involutive algebra
A
0
0
is a C*-algebra.
The question of quotient C*-algebras is a more delicate matter (cf.
1.8.2).
9
1.3.4.
IIxl1
2
IIx*II’llxll, hence
1.3.5.
Let
For each x
[z] = sup [xx’[,
Ilx’lI"’l
In fact, it is clear that Ilx’llos; 1 implies Ilxx’ll OS;
[x], To show that
sup [xx’]~ Ilxx*11 = IIxl1
2
=1.
11111
2
11111 =
0 or 1.
We thus see that, unless A = 0, 11111 = I, and it follows that if A,t 0
then
1.3.7. We recall that no continuity
condition
the definition of
morphisms of normed involutive algebras. We shall see that, for C*›
algebras, a morphism is automatically continuous, or, more precisely,
that we have:
PROPOSITION. Let A be an involutive Banach algebra, B a C*-algebra
and 1T a morphism of A into B. Then 1I1T(x)1I
os; [x] for every x E A.
For each hermitian element y of B, we have Illll = Ily*yll = IIyl12 and
hence by induction IIl
As n --,> +00 the left hand side of this
equation tends to the spectral radius p(y) of y (cf. Bl), so that
(1)
Sp ~1T(X) ~ Sp ~x so that p( 1T(X»
os;
1/1T(X)W =111T(x*x)1I =p(1T(X*X)) ~
lIx*xll
IlxW•
1.3.8. The following proposition will enable us almost always to confine
our attention to unital C*-algebraswithout any
loss of generality.
PROPOSITION. Let A be a C*-algebra and A the involutive algebra
obtained from A by the adjunction of an identity. Then the norm on A
can be extended to A in exactly one way that makes A a C*-algebra.
The uniqueness follows from 1.3.7. We
prove
e and let 1
complementary self-adjoint two-sided ideals, and so
there is an isomorphism of the involutive algebra A onto the involutive
algebra C x A
C x A
that
an
identity. Each element x of A defines an operator of left multiplication
L, in the two-sided ideal A of A; put [x] = IILxll. I f x E A, this coincides
with the original norm on A, by 1.3.5. Moreover, it is clear that x ~ Ilxll is
a seminorm on A and
that Ilxyll ~ Ilxll•llyll. This seminorm is in fact a
norm, for le t x = ,\ - x’ (,\ E C, x’ E A) be an element of A such
that
lIL
x
l1=0, i.e. such that xy =0 for each yEA. We show that then x =O. I f
,\ ¥ 0, we have for each yEA,
0 = ,\-I
-lx’y, ,\ -IX’ is a left
identity for A, and A possesses an identity (1.1.5) contrary to hypo›
thesis. Hence ,\ =0, so
0 implies
and of
codimension 1 in A, A is also complete. I t remains to show (1.3.4) that
IIxl1
2
and it is enough to do this for IIxll
= 1. For
each r < 1, there is yEA such that Ilyll ~ 1 and IIxyll2 ~ r; then, as xy E A,
we have Ilx*xll ~ Ily*(x*x)YII =II(xy)*xYII =IIxyl12 ~ r and therefore Ilx*xll ~
1.
(i)
If h is a hermitian element of A, each element of Sp’h is real.
(ii) If
A has an identity, and u is a unitary element of A, each element
of Sp u has absolute value 1.
In
1.3.8,
that A
has an identiy (and that A¥ 0). We have Ilull = Ilu-111 =1 by 1.3.6, so that
p(u) ~ 1, p(u-
in the
 
2! 2!
= exp(- ih),
thus
and so
z E R. (Throughout the book, R denotes the set of
real numbers.)
1.3.10. The following proposition shows that, for C*-algebras, it does
not
matter
which
PROPOSITION.
Let A be a C*-alge ra, B a sub-C*-algebra and x an
element of B. Then
(i) Sp~x = Sp~x.
(ii) If A has an identity which also lies in B, then SpAx = SPBx.
We see
(i) follows from (ii) on adjoining an identity . We therefore
prove (ii). I f x is hermitian, we have
SPBX C R (1.3.9), so that SpBx =
SpAx (B 2). In
the
general case, if x E B is invertible in A, xx* is also
invertible in A,
and so x has a right inverse
in
B;
similarly
x
in B and is therefore invertible in B.
Applying this result to x - A . 1 where A E C,
we obtain (ii).
C*-algebras are called completely regular algebras in [108] and B*›
algebras by numerous authors.
1.4. Commutative C*-algebras
the
be a commutative C*-algebra, S its spectrum (which
is a locally compact space), and B the C*-algebra of
continuous
(i) Every character of A
is
hermitian.
(ii) The Gelfand map is an isomorphism of the C*-algebra A onto the
C*-algebra B.
is an hermitian element of A, we have
 
x(x*) =
which proves (i).
=
and
for
each
point of S there is at least one function belonging to g;(A) which does
not vanish there (B 3).
The
will be complete if we now show that fJ’ is
isometric. Now, 1IfJ’(Y)!1 =!Iy!l for
any hermitian y [1.3.7, formula
(1)],
and
IIfJ’xI1
fEB ,
Sp’f
it
is possible to define hex) E A for any continuous complex valued h
defined on
any,
riot
useful tool in the sequel.
1.4.3. PROPOSITION. Let
spectrum and let x
E A. Suppose that the sub-C*-algebra of A generated
by
1
A.
homeomorphism
of
Santo
SPAx.
This map is continuous and its range is SPAX (B 3). Moreover, let X,
X’ E
X
and
X’
A’ =
under
consideration is injective, and is therefore a homeomorphism as S is
compact.
1.5. Functional calculus in C*-algebras
1.5.1. THEOREM.
Let
A be a unital C*-algebra, x a normal element of A,
S SpAx and A’ the C*-algebra func›
tions on S. Then there is a unique morphism f/> of
A’ into A such that
 
13
~
</>(A’)
is the sub-C*-algebra of
A generated by 1 and x and therefore consists entirely of normal
elements.
The
into
A
the
spectrum, C the C*-algebra of continuous complex-valued functions on
T and :JiB
B
Tonto
SPBX
A’
~ C
that
the composite isomorphism
A'~C~B
Composing this with the canonical injection of B into A, we obtain a
morphism of
1.5.2. DEFINITION.
Let A
be a unital C*-algebra. I f x is a normal
element of A and if f is a continuous complex valued function ’on Sp
AX,
The
fact
expressed
continuous complex-valued
in z and Z,
then f(x) = P(x, x*) where Ptx, x*) has its usual algebraic interpretation
(remember that xx* = x*x).
With the above notation, we have
or in
[CH. I, §5
1.5.3. PROPOSITION. Let A andB be unital C*-algebras, 4> a morphism ofA
into B mapping 1 to 1 and x a normal element ofA o that q,(x) is a normal
element ofB. Let f be a
continuous complex-valued function on SPAx. Then
if the restriction
of
f to SPB4>(X) is again d noted by f, we have
4>(f(x»
=f(4)(x)).
Let C be the C*-algebr of continuous complex-valued functions on
Sp
are morphisms of C into B
which take the same value when f is anyone of the functions z ~ I,
z ~ z, z ~ i. Since the sub-C*-algebra of C generated by these functions
is equal to the whole of C, th two morphisms
are
identical.
1.5.4.
element of A, [fI the Gelfand map for
A,
[fl.
1.5.5.
COROLLARY. Let A be a unital C*-algebra, x a normal element of
A, C the C*-algebra of
continuous complex-valued functions on Sp x, f
an element of C,
functions o Sp f(x) =f(Sp x),
and
is a morphism of
into A which maps 1 to 1
and the function z ~ z to f(x). From the uniqueness statement of
theorem 1.5.1, it follows that (g 0 f)(x) =g(f(x».
1.5.6. PROPOSITION. Let A be a C*-algebra, x a normal element
of A,
S A’ the C*-algebra continuous complex-valued functions
on S which vanish at o. Then there is exactly one morphism q, of A’ into
A such that q,(L)=x where Lis the function
z
on S. This morphism is
isometric and its image 4>(A’) is the sub-C*-algebra of A generated by x
which therefore consists entirely
dense
in
A’, the uniqueness of q, is immediate. The existence follows from
theorem 1.5.1 on adjoining an identity to A.
1.5.7. By the adjunction of an identity, all the results of this section can
be extended immediately, with obvious modifications, to the
case
of
non-unital C*-algebras. We shall thus make use of these results even
when an identity is not assumed to be present.
 
15
element
the
variable
Ixl=!J(x). These are hermitian elements of
A, (and, indeed, of the sub-C*-algebra of A generated by x), because
II =
=fz,
13 =!J. Since i.. t-. f3 take non-negative values only, we
have
(1)
(2) x =x ’ - x", [x] =x" +x", x+x- =x-x+ =
O.
The norm of Ixl is the same as that of x,
while the norms of x+
and
x›
to that of
x. We note, with a view to no. 1.6.4, that
x" and x- are
product
is
fz.
1.5.8. Let A be a C*-algebra. For each positive integer n, we have
A =An, the set of linear combinations of products of n elements of
A;
it
that
each hermitian element x of A is a product of n
elements of A. Now, if
t.. .. .i, are continuous real-valued functions of
a real variable such that
then
1.6.1.
of
(i) Sp~~ O.
(ii) x is
(iii) x is of
 
ALGEBRAS [CH. I, §6
Furthermore, the set P of those elements which satisfy these conditions
is
P
condition (i).
of

(iii) ~ (i): if x = h 2 with h we have Sp ~x = (Sp ~h)2 ’""0
Sp
~h
and the last part of the proposition
we need the following lemmas:
1.6.2. LEMMA. Suppose that A is unital. If x
E
lit - x]
~
of
A
enough
to
consider
be
reduced
to
the
case
where
complex-valued functions on a
1.6.3.
is
unital and let x be n hermitian
element of A. In order that x E P, it is necessary and sufficient that
II(IIxll
then follows at once from Lemma 1.6.2.
1.6.4. We now return
closed convex
can assume. thanks
clear
Ax E P. We now show that x +yEP
whenever x,
~ 1. Then III - x]
and thus x + yEP . I f
x E P
n ( -P) then Sp x = {O}, p(x) = 0 and so x = 0 [1.3.7, formula (1)].
We
finally
0 (1.5.7).
si ce (iii)
= -(vy)(vy)*+2s
(vy)(vy)* E P
because, in any algebra, the spectrum of a product is
inde endent of the order of its factors (B 26). Hence (vy)(vy)* E
P
4
now proved.
1.6.5.
DEFINITION.
Let A be a C*-algebra. An element x E A is said to
be
positive
x ~ 0 if it is hermitian and satisfies the three
equivalent conditions of proposition 1.6.1. The set of positive elements
of A is denoted by A ".
I f B is a sub-C*-algebraof A and
x E
statement x ~
0 has the same meaning whether it is interpreted in A or in
B.
(-A +) ={O},
the relation x - y ~ 0
is a partial ordering in A which is compatible with the real vector space
structure of A; we write this relation x ~ y or y ... x. I f z is an hermitian
element of A then z ’ ~ 0, z: ~
0 and [z] ~ 0 by 1.5.7, formulae (I); z"
(resp. z ") is a led the positive (resp. negative) part of z. By 1.5.7,
formulae (2), each hermitian elemen of A is the difference of two
elements of A ",
1.6. . Let A be the C*-algebra of continuous complex-valued functions
vanishing at infinity on a locally compact space
T
and suppose f E A.
It follows from condition (I) of proposition 1.6.1 and the fact that
Sp ~f = f( T) U {O} that the relation f ~ 0 has its usual meaning in the
algebra A.
1.6.7. Let H be a Hilbert space A the C*-algebra 2(H), and x an
element of
0 is equivalent to the
condition (xg Ig)
~ 0 for every g E H, i.e. to the usual definition of
positive operators. If x = y*y for some yEA , we have
(xg Ig) = (y*yg
 
E H. For every TJ E
H,
we
have
TJ
b,
a
:s;:; b, then
x*ax:s;:; x*bx, as there is yE A such that b - a = y*y from which it
follows that
O.
Now suppose
that A
is unital and let b be an element of A with b ~
1
(hence b is invertible). Applying the above with x =b-
l12
1 ~
b
-I . More generally, let a and b be invertible elements of A + such
that O:s;:;
1:s;:;a-
1I2ba-
1I2
1.6.9.
be such that y :s;:; x. Then
Ilyll :s;:;[x]. In fact, we can assume
A
y :s;:;
a morphism of A into B.
I t is plain that
c/>(A+)
C
</>(A)
n
E
</>(A)
E
2
E </>(A
References: [604], [918], [1101], [1323], [1477].
1.7. Approximate identities in C*-algebras
1.7.1. Let A be a C*-algebra. We say that an approximate identity (uJ
of A (B 29) is increasing if U
A
JL implies
uA:s;:; U
w
1.7.2. PROPOSITION. Let A be a C*-algebra, and m a two-sided ideal o f
A which is dense in A.
Then
o f
separable, this approximate
Let A
be the C*-algebra obtained by adjoining an identity to A. Let
A
=
A
t ;;?;0, we have o:s; U
A
;=1 n n
Now the function of a real variable t ~ t(lln +t)-2 is always :s; ~n. Thus
1
L
[(u
A
A
that
I
[(u
A
- 1)x;][(u
A
- l)x;]*:S;-,
4n
)
is thus an approximate identity. Now let A, p., E A
be
such
that
p.,
={XI,
VA :s; VI" and (lIn + VA)-I;;?; (lIn +VIL)-I by 1.6.8.
For
1(1
)-1
1(1
)-1
1(1
)-1
 
. . .
)
for
each
deduce
that
and
same
way.
1.7.3. Let A be a C*-algeb aand I a right ideal of A. Then
there exists a
family (uJ in
I n A +
that (I) IluA11 ~ 1;
of 1.7.2
1.8.1.
PROPOSITION. Let A be a C*-algebra, B a normed involutive
algebra and
B. Then
Let x E
A. I f we can show that 111J(x*x)11 ~ Ilx*xll, we can
deduce
that
IIxl1
2
assume
A
can
assume B is commutative, and we then replace B by its
completion.
Furthermore,
A
and
B.
In short
we may confine attention to the case in which A and are commutative,
complete and unital.
are compact spaces. For each X E T,
X 0 1J is a
character
denote
by 1J’(X); if x E A. then 1J’(X)(x) =
X(1J(x))
x; 1J’
T
f#
0 and f 11J’(T) =
O. By 1.4.1. is the Gelfand transform of an E We
have
1.8.2.
PROPOSITION. Let A be a C*-algebra and I a closed two-sided
ideal of A. Then I is self-adjoint and AII. endowed with the natural
 
21
Let (UA) be a family with the properties of 1.7.3. I f x E I, we have
and xr u, E
I, so that x*
We know that AI
a Banach algebra. Denote by x ~
x
onto AII. To
show that AII is a *-algebra, it is enough to show that Ilxll
2
(1)
In fact, if Y E I we have /lAY - Y ~ 0, and so
lim Ilx - uAxil =
lim [x - UAX +
(we are working in ,4). Thus
Ilxll?; lim [x -
yEI
which proves (1). This established, we have for each Z E
I,
A
+ UAZU
A
,,;::;; Ilxx*ll.
1.8.3. COROLLARY. Let A and B be C*-algebras, cP a morphism of A
into B
cP:
I is in cP(A) is in Band IjJ is an
isomorphism of the
cP(A).
Since cP is continuous (1.3.7), I is closed and AII is a C*-algebra
 
quotient is injective and therefore isometric (1.3.7 and 1.8.1). Hence
et>(A)
is complete
1.8.4. COROLLARY. Let
A be a C*-algebra, B a sub-C*-algebra of A
and I a clo ed two-sided ideal of
A.
and the C*-algebras (B +1)1 I and
BIB n I are canonically isomorphic.
Let et>
be the canonical morphism A ~ AII, and let I/! be the restriction
of et> to B.
Then I/!(B) =(B +1)11 is closed in A ll (1.8.3), and so B +I is
closed in A. Consider the canonical decomposition of I/!:
B
~ BI(B n I)~ I/!(B)~ All.
By (1.8.3), the morphism BI(B n I)~ l/J(B) =(B +1)11 is a C*-algebra
isomorphism.
A be a C*-algebra, I a closed two-sided ideal
of A and J a closed two-sided ideal of I. Then J is a closed two-sided
ideal of A.
= AJ3A c III c J.
References: [618], [893], [1101], [1323], [1456]. The proofs of 1.8.2 and
1.7.3 were
communicated to me orally by F. Combes.
1.9.1. Let A be a C*-algebra. I f the norm and all the algebraic
operations of
A are retained with the exception of (A, x)~Ax which is
changed to (A, x)
~
Ax, a C*-algebra A called the conjugate of A is
obtained.
1.9.2. Let A be a C*-algebra. I f x E A is normal, then [x] =p(x). (Use
1.4.1) [1323].
1.9.3. Let
A be a C*-algeb a.
(a) I f every maximal commutative sub-C*-algebra of A h s an iden›
tity, so does A.
mensional, so is A. [1159].
1.9.4. Let
A
be a C*-algebra. I f the conditions xEA+, yEA+, x~y
imply x
 
ALlDENDA
23
*1.9.5. Let A be a unital Banach algebra endowed with an involution
such that Ilx*xll =Ilx*II’llxll
for each
x E
A. Then
[1187].
1.9.6. Problem: does a unital C*-algebra A of dimension >1, whose
only closed two-sided ideals are 0 and. A possess any projections other
than 0 and I? [903].
1.9.7.
be a unital C*-algebra, and x an element of
A
Then x*x is not invertible in the sub-C*-algebra
B generated by I and x*x, so there exist Yt, Y2, . . . E B such that IIYnl1 = 1
and IIYnx*x]
E
topological divisor of zero. [1323].
1.9.8. Let
A
be a unital C*-algebra, N the set of normal elements of
A,
x
E
a neighbourhood of 0 in C. There exists a neighbourhood
U of x in N such that for every Y E U, Sp Y C Sp x +V and Sp x C
Sp Y + V. [1323].
*1.9.9. Let A be a unital C*-algebra, H a Hilbert space and c/J a linear
map of A into 2(H) such
that c/J(A+)
2
(a)
Let
p:
=
p(x
n
) =p(x)" for X hermitian and n an integer >0. Then p(A+) =B+, p
is isometric, p maps a pair of commuting elements into a pair of
commuting elements, and pO) = 1.
(b) Let (T: A
~
B be an isometric bijective linear map. Then (T is he
composition of a map having the properties of (a) and multiplication on
the left by the element (T(l) of B.
(c) Let T: A
~
B be an isometric linear map such that T(l) = 1. Then
r(x*) =T(X)* for each x E
A. [179], [837].
(b) Let D be a derivation
of A and x a normal elemen of A. I f
x(Dx)
=(Dx)x, then Dx =O. In particular, the only derivation of a
commutative C*-algebra is the zero operator. It follows from this that a
commutative closed two-sided ideal in a C*-algebra is central.
(c) Let D be a derivation of A and
x
Dx
A
and
I
Then D(I)
C I (use 1.5.8).
(e) Let H be an infinite-dimensional Hilbert space, A the C*-algebra
of all compact operators in H and x an element of
.2(H)
xy - yx of A is not inner.
[530J, [532], [535J, [538], [1416J, [1417], [1428],
[1429J,
[1433].
(a)
A.
Then the product
ideal IJ is equal to I n 1. (If x E (I n I t ,
then X
Further,
(I +It=I"+I", [1239], [1515].
(b) I f I n 1= 0, the canonical map I +I ~ I x I is a C*-algebra
isomorphism.
A
whose
a
All; (resp. A/I). For each E A,
Ilw(x)11
= [584).
1.9.13. Let A be a C*-algebra. A C*-semi-norm on A is a semi-norm N
such that
N(x):s;; Ilxll,
N(xy):s;; N(x)N(y) ,
N(x*x) = N(X)2
for any x, yEA. The set X(A) of all C*-semi-norms on
A
is compact for
the topology of pointwise convergence on A. I f I is a closed two-sided
ideal of A and x E A, let N1(x) be the norm of the canonical image of x
inA/I. Then I~NI is a bijection of the set ~(A) of all closed two-sided
ideals of onto X(A). I f I,
I E ~(A), we have
N
InJ
sup(N
b
N
J
) [use
1.9.12.b]. A C*-semi-norm N is said to be extremal if N cannot be the
upper bound o two C*-semi-norms without being equal to one of them.
For a C*-semi-norm N to be n n-z ro and extremal, it is necessary and
sufficient that I be prime. [In any algebraR, a two-sided ideal I is said to
be prime if l’i’R and if the relation J’J" c I, where J’, J" are two-sided
ideals of implies
that I’c I or J" C I.] I f is separable, the set of
extremal C*-semi-norms is a Os in X(A). [The set of (N, N’) E X(A) x
X(A) such that N:s;;N’ or N’:s;;N is compact. Its complement is a
countable union of compact sets, and its image in X(A) under the
continuous map (N,
[584].
1.9.14. Let (Ai)iEI be a family of C*-algebras. Let A be the set of
x
ADDENDA 25
number of indices i. I f we put [x] = supllx], A becomes a C*-algebra in
a natural way, and this algebrais called the restrictedproduct of the Ai’
1.9.15.
The
radical R of a C*-algebra is zero. [If x E R, then 1+Ax*x is
invertible in A for
 
One of the most important concepts of this book is
that
of a represen›
tation. Given an involutive Banach algebra A, it would be a difficult task
to establish the existence of representations of A directly. However, we
shall set up a correspondence between representations of A and positive
forms on A, and in particular between irreducible representation and
pure positive forms. Moreover, the classical tools of functional analysis,
namely the Hahn-Banach and Krein-Milman theorems, enable us to
prove the existence of positive forms and indeed of pure positive forms
as well. This is the basic idea of this chapter.
In the various results which fo low, the involutive algebras being
studied are subjected to a variety of conditions. At a first reading,
however, it may be assumed that we are concerned exclusively with
C*-algebras.
DEFINITION. Let
A be an involutive algebra. A linear form f on A
is said to be positive if
f(x*x)
~ 0 for each x E A. I f A is a
normed involutive algebra, a state of A is a continuous positive linear
form f on A such that 11111 = 1.
Let A be an involutive algebraand f and g linear forms on A. We say
that f dominates g and we write f ~ g or g :s;; f
if
f - g is positive. This
defines a preorder in the dual of A which is compatible with the real
vector space structure. I f
A is a C*-algebra this relation is in fact a
partial ordering, for if f ~ 0 and f:S;; 0, then f
vanishes on A
+, therefore
on the set of hermit an elements of A (1.6.5) and hence f =
O.
Let a be a locally compact space and A the C*-algebra of continuous
complex-valued functions on a which vanish at infinity. A continuous
linear form on A is simply a boun ed measure f..t on a, and to say that
this linear form is positive is exactly the same as saying that the measure
f..t is positive.
f
x,
yEA put (x I
y) = f(y*x). T is scalar product is linear in x, anti-linear
in and (x Ix)
~ 0 for each x. A is thus endowed with a pre-Hilbert
space structure.
A,
yEA) ,
I f ( y*xW~f ( x*x ) f ( y*y ) (x E A,
yEA) .
I f A is unital, we deduce, putting y =1 in (1) and (2), that
(3)
(4)
from the pre-Hilbert space A, so that H
=
AIN where N is the set of
those x E A for which f(x*x) =O. By (2), N is equaIly the set of x E A
such that f(yx) = 0 for all yEA , so that N is a left
ideal of A.
[Putting (x Iy) = f(xy*) we would obtain another pre-Hilbert space
structure on
unital Banach
with Ilx’ll ~ 1 and x = I +x’. The series
converges to an element y of A such that i = x. If A has an isometric
involution and if x is
hermitian, then
+...
+ -
G-n +1) " l I
+. ..
is convergent rom which it foIlows that the element y exists. I f we
compute i, we obtain a power series in x’ whose coefficients we
 
POS IT IVE FORMS AND REPRESENTAT ION [CH. 2, §1
l = 1 x’=
x. I f A hasan isometric involution and if x is hermitian, y is
seen to be a limit of hermitian
elements and is therefore hermitian
because the involution is continuous.
2.1.4. PROPOSITION. Let A
that
/11/1 =1. I f f is a positive linear form on A, then f is
continuous and IIfli = f(1).
and Ilxll :;;;
then 1 - x an be written in the form
y*y (lemma 2.1.3), from which it follows that f (1- x) ~ 0 and so tu:«
f(I).
and IIx
and hence, using 2.1.2,
This
«i:«11111, and hence 11111 = f(1)•
2.1.5. PROPOSITION. Let A be an involutive Banach algebra having an
approximate identity (B 29), A the
involutive
algebra
obtained from
A by
adjoining an identity to it, and f a continuous positive linear
form
If(x)1
2
(ii)
(iv) If (Xi)iEI is a family o f
elements
of
A
approximate identity
f(u~u)~II11I·
A
form
space
A
f(x*) = lim f(x*u
If(x)1
2
This
proves (i) and it is plain that (i) :? (iv) :? (iii). We prove (vi). The
uniqueness of
of
+
x of A (A E C, x E A) put /(A +
x) =
29
,\11111 +f(x): Then f is a linear form on A extending f and using (i), we
have
1,\1
1,\1
1
and so f is positive. Clearly, j(1) = 11111. Now let
g
be given an involutive Banach algebra
structure extending that of A and such that 11111
= 1. By
g«’\+x)*(,\ +x) =
11111
= f«,\ +x)*(,\ +xj),
so that g ~ f, and (vi) is proved. I f yE A, the form x --+ j(y*xy) on A is
positive, because j ( y* (x*x )y )=f«xy )* (xy»~O; by (2.1.4), its norm is
f(y*y) and (ii) now follows.
In the notation of (iv), we have
f«x; -
1)*(Xi -I)
(ujluj)=f(u~uj),,;;;III1I. For every xEA , (ujX)=f(x*uJ--+
f(x*) =(11 x); since A is dense in A (d. (vii», we see that u, converges
weakly to 1. Hence f(u;) = (u, 11)--+(1, 1) =
/(1)
2.1.5, f is called the canonical extension of f to
A.
COROLLARY. Let f, g be continuous positive forms on A and
f,
II!+gil
g.
The first relation follows from 2.1.5 (v), while the second follows from
the first.
I t follows that the set of states of A is a convex subset of the dual of
A.
[CH. 2, §l
2.1.7: We retain the notation of 2.1.5. Now let Q be the set of con›
tinuous positive forms on A and Q the set of positive forms on A such
that g(1) =
Ilg IA]. Then 1....0,1is a bijection of Q onto Q which preserves
both the additive and order structures. I f a positive form on A is
dominated by an element of Q, it must itself belong to Q; in fact let
g =gl g2 where g E Q and gl, s. are posi ive forms on A, and let I, II,
12 be the restrictions of g, gt, g2 to A. We have
s, +
and thus gl E Q, g2E Q.
2.1.8. Let A be a C*-algebra and I a positive form on
A. Then I is

that Ilx;11
I.
We show that the numbers I(x;) form a bounded set. For
(AI> A
converges to an element X of A. For
every integer n ~ 1, we have L7=n+1 AjXj ~o by 1.6.1, and so
Hence L7~1 AJ(x;) <+00 . Since this holds for any sequence (A;) of
non-negative reals with
L7~1 A <+
numbers
I(xj)
M= sup I(x)<+oo.
XEA+.llxll"’l
I f x is hermitian and f norm ~ 1, we have
and if x is any element of norm ~ 1, then
Hence I is continuous.
REPRESENTATIONS
31
In the sequel, positive forms on C*-algebras will be mentioned over
and over again, and the reader should never forget that these forms are
automatically continuous.
2.1.9. PROPOSITION. Let A be a unital C*-algebra and I a continuous
linear [orm on A. For I
to be positive, it is necessary and sufficient that
Ilfll=1(1)•
I f I ’;if:0, then Ilfll=1(1) (2.1.4). Now suppose Ilfll=I(l). Let x E A
+ and
;;!:
0. (We can, of course, assume that 1(1) = 1.) There is
a closed disc [z - zol-s; p in C which contains Sp x but not I(x). Now the
spectrum of the normal element x - Zo is contained in he disc Izl-s; p,
from which it follows that Ilx - zoll-s; p. Hence
’/(x) - zoll = I/(x) - zof(1)1
=I/(x - zo)l-s;
IIfll•llx
- zJI ’;if:p,
whi h is a contradiction. (An alternative proof is based on B 28.)
The
References: [174], [618], [619], [638], [1097], [1101], [1323], [1455].The
result of 2.1.8 can be generalized to involutive Banach algebras with an
approximate identity [1755].
2.2. Representations
2.2.1. DEFINITION. Let A be an involutive algebra and H a Hilbert
space. A representation of A in H is a morphism of the involutive
algebra A into the involutive algebra X(H).
In other words, a representation of A in H is a map ’TT’ of A into 2(H)
such that
’TT’(xy) = ’TT’(x )’TT’(y),
’TT’(Ax) =A’TT’(x),
’TT’(x*)=’TT’(x)*
’TT’
’TT’.
The space H is called the pace of ’TT’ and is denoted
by H".
A
are said to be
equivalent, and we write ’TT’ = ’TT" if there is an isomorphism U of the
Hilbert space H onto the Hilbert space H’ which transforms ’TT’(x) into
’TT"(x) for
each x
class 01 representations.
(For convenience, we will often not distinguish between representations
and classes of representations.)
2.2.2. An intertwining operator
= 7T’(x)T for any x E
A.
The
operator U of the previous definition is an intertwining operator. The set
of all intertwining operators for 7T and 7T’ is a vector space whose
dimension is called the intertwining number of 7T and 7T’, and is the same
as the in ertwining number of 7T’ and 7T.
For let T: H ~ H’ be an
intertwining operator for ’TT and 7T’. Then T*: H'~H is an intertwining
operator for 7T’ and 7T, because
T*7T’(X)= (7T’(X*)T)* = (T7T(X*))* = 7T(x)T*.
It follows from this, that
T*T7T(X) = T*7T’(x)T =’7T(x)T*T,
and thus ITI =(T* T)1/2 commutes with 7T(A). Let T = U TI be the polar
decomposition of T. We have, for each x E A,
(1) U7T(x)IT! =UITI7T(X) = T7T(X) = ’TT’(x)T= 7T’(x)UITI.
I f Ker T
(2)
U7T(X) =’TT’(x)U.
I f moreover T(H) =H’, i.e. if Ker T* =0, then U is an is morphism of
H onto H’, and (2) shows that
7T and 7T’ are equivalent.
2.2.3. Let (7TJiEI be a family of representations of A in Hilbert spaces
Hi’ Let H be the Hilbert sum of the Hi’
I f the set of numbers II’TTi(x)1I is
bounded for each x E A (which is th case, by 1.3.7, if A is an involutive
Banach algebra), we can construct the continuous linear operator’TT(x) in
H which induces 7Ti(X) in each Hi’ Then x ~ ’TT(x) is a representation of
A in H, known as the Hilbert sum of the
7Ti
7T1 E9 7TzE9’
in the case of a (finite) family (7Tl’ 7Tz, . . . , 7T
n
) of
representations.
I f
(7Ti)iEI is a family of representations of A each of which is equal to
7T, 1= E97Ti
denoted by C7T. Every representation equivalent to a representation of
this type is called a multiple of
’TT.
2.2.4. Let 7T be a representation of A in H.
I f a closed vector subspace
K of H is invariant under 7T(A), the restrictions of the 7T(X)’S to K
define a subrepresentation tr’ of A in K denoted by TTK or 7TE if E =P
K
is also invariant under 7T(A), since if g E
K
and
"1 E H e K, we have that 7T(X*)gE K for each x E A, and therefore
(7T(x)T/1 g)=(T/I7T(X)*g) =(T/I7T(X*)g) =0,
so that 7T(X)T/ E He K. Hence P
K
commutes with 7T(A). I f tr" is the
subrepresentation of 7T defined by He K, then 7T = 7T ’ E!1Tr".
Let
p, p’ be two representations of A. I f p’ is equivalent to a
subrepresentation of p, we say that p’ is contained in p or that p
contains o’, and we write p’ ~ p or p.~ p’,
2.2.5.
The closure of
7T(A)g is a closed subspace of H, invariant under 7T(A). I f this subspace
is equal to H, we say that
g
is a cyclic vector for 7T.
2.2.6. PROPOSITION. Let Tr be a representation of A in H,
and
let K be
the closed subspace of H generated by the 7T(X)g (x E A, gE H). Let K’
be the closed subspace of H consisting of those gE H such that 7T(X)g=
o for every x E A. Then K and K’ are invariant under 7T(A)
and
7T(A)’
[the
direct sum is equal to H.
Clearly, K and K’ are invariant under 7T(A). Any operator that
commutes with 7T(X) leaves the kernel and range of 7T(X) invariant so
that K and K’ are invariant under 7T(A)’.
Let
hence K ’=HeK .
K is called the essential ubspace of 7T. 7T is said to be non-degenerate
if K
=H. The above argument shows that every representation of A can
be uniquely expressed as the direct sum of a trivial representation and a
non-degenerate representation.
2.2.7. I t is plain that a direct sum of non-degenerate representations is
non-degenerate and that a representation that admits a cyclic vector is
non-degenerate. Conversely:
PROPOSITION. Every non-degenerate representation
Let
t t
be a non-degenerate representation of A in H with H ¥-O. It is
enough to show that there is a sub-representation of
Tr
in a non-trivial
space which admits a cyclic vector (for the result then follows from an
application of Zorn’s lemma). Let g be a non-zero element of H. The
closure K of Tr(A)g is non-trivial and invariant under 1r.
Let
L =
He K, and put g=g! +g2 with gl E K, g2 E L. I f x E A, then 1r(x)gt E
K,
1r(x)gl +Tr(X)g2 =1r(x)g E K, so that Tr(X)g2 =0; con›
sequently Tr(x)g =1r(x)g!. Thus K is the closure of 1r(A)g!.
2.2.8. Let t t be a representation of A in H and let i i be the Hilbert
space conjugate to H (ii=H, ut when passing from H to ii, mul›
tiplication by A E C is replaced by multiplication by
A
AO
be the
reversed involutive algebra of A. It is easily checked that the map
x ~ 1r(x*) is a representation of AO in tt, which is denoted by iTo.
2.2.9. Let A be the involutive algebra obtained from A by adjoining an
identity and let 1r be a representation of A in H. Then t t has a unique
extension (known as the canonical extension) to a representation i i of A
in H such that i i ( l )=1.
2.2.10. Let
A be an involutive Banach algebraand t t a representationof
A in H. Recall (1.3.7) that 1I1r(x)11 ~ Ilxll
for every x E A. I f A has an
approximate identity (u
is nondegenerate, then 1r(uJ tends
strongly o 1. Indeed, for eac yEA and each gE H, Tr(uJ( Tr(Y )g) =
Tr(UiY)g tends strongly to 1r(y)g since
IITr(u;y) - Tr(Y )11 ~ IluiY - yll ~ 0;
now, the Tr(y)g constitute a total subset of H and moreover 111r(uJII ~
lIu;11 ~ I, so that 1r(uJ1/ tends strongly to 1/ for each 1/ E H.
References: [618], [1101], [1323].
2.3. Topologically irreducible representations
A be an involutive algebra, H a Hilbert space
and t t a representation of A in H. Then the following conditions are
equivalent:
35
(i) The only closed subspaces ofH invariant under 7T(A) are 0 and H;
(ii) The commutant
of 7T(A) in 2(H) is just the set of scalar operators;
(iii) Every non-zero
vector of H is a cyclic vector for 7T, or 7T is
I-dimensional.
Let
~ E H, f¥- O. I f 7T (A )~
is not dense in H, then 7T(A)~ =0 by (i). Thus C~ is invariant under
7T(A), a d so H = C~ and 7T is I-dimensional.
(iii)::} (i): suppose condition (iii) is satisfied.
Let
7T(A).
=
1. Now suppose that every non-zero vector of H is a
cyclic vector f r 7T, and let ~ E K, ~¥ - O. Then 7T(A)~ C K and 7T(A)~=
H, so that K
H.
(ii)::} (i): suppose condition (ii) is satisfied. Let K be a closed sub›
space of H invariant under 7T(A). P
K
and is therefore a scalar operator so that either P
K
= 1, i.e.
K=OorH.
(i)::} (ii): suppose condition (i) is satisfied. Let T be an element of
2(H)
T +T*
and T - T* commute with 7T(A), we need only consider the case of
hermitian T. The
7T(A)
and
are all therefore equal to either 0 or 1, by (i). Hence T is scalar.
2.3.2. DEFINITION.
and
7T
7T
irreducible if H
¥- 0 and if 7T satisfies the equivalent conditions of 2.3.1.
Such a representation is either trivial, l-dirnensional or else non›
trivial and non-degenerate. We denote by A the set of classes of non-trivial
topologically irreducible representations of A.
Remember that irreducibility of 7T in the algebraic sense means that
the only subspaces of H invariant under 7T(A)
are {O} and H. I f dim H =
+00, this is a condition far more restrictive than topological irreducibility.
We shall presently see (2.8.4), however, that if A is a C*-algebrathen the
two notions are equivalent
A finite-dimensional representation of A is a direct sum of topologi›
cally irreducible representations (2.3.5). We shall see (8.5.2) that this
result also holds to a ertain extent for infinite-dimensional represen›
tations, albeit in a rather more subtle form, which is one of the reasons
why we shall be particularly concerned with topologically irreducible
 
representations in what follows. A fundamental problem associated with
any given involutive algebraA is that of determining, up to equivalence,
all the topologically irreducible representations of A. I t should be noted
that there’may well not exist any non-zero topologically irreducible
representations(or even any non-zero representationsat all) of A; we shall
nevertheless see (2.7.3) that every C*-algebrahas "enough" topologically
irreducible representations.
2.3.3. Let A be a separable normed involutive algebra and 7T a to›
pologically irreducible representationof
be a non-zero vector of H; if (x
n
) is a dense sequence in A, then the
7T(Xn)~ are dense in H. The same argument shows that, for any normed
involutive algebra B, the dimensions of the topologically irreducible
representations of B are bounded above by some fixed cardi al.
2.3.4. Let
A be an involutive algebra and 7T, 7T’ two topologically
irreducible representations of
I f
n = 0,
then 7T and 7T’ are not equivalent. I f n > 0, let T: H",~ H"" be a non-zero
intertwining operator. By 2.2.2, T*T and TT* are non-zero scalar
operators and 7T and 7T’ are equivalent.
2.3.5. Let A be an involutive algebra. We study the finite-dimensional
representations of A, and while what we say here can be regarded as a
special case of certain theorems which we shall meet later on, or equally
as a special case of some purely algebraic results, we think it better, f r
the convenience of the reader, to present a direct treatment here and
now.
7T = 7TI EB .. . EB 7T
n
,
where the 7Tj are irreducible. This is obvious if dim 7T
=0 with of course,
n =O. Now suppose that dim 7T =q and that the assertion has been
proved for dim 7T
<
q. I f 7T is irreducible, there i nothing to prove.
Otherwise, 7T =7T’ EB 7T" where dim 7T’ < q,
dim 7T" q and we merely
have to make use of the inductiv hypothesis. The decomposition
7T =7T1EB .. . EB 7T
~
en
uniqueness result. Let PI and P2
be two irreducible subrepresentations of
7T, and PI and P
2
REPRESENTATIONS
37
projections commute with ’IT(A), and so the restriction of P2 to HpJ is an
intertwining operator for PI and P2’ Hence, unless H
p1
orthogonal, PI =" P2 (2.3.4). This proves that every irreducible subre›
presentation of ’IT is equivalent to one of the ’IT/s, and thus, rearranging
the ’IT/s, we see that ’IT = VI EB ... EB V
m,
where each Vi is a multiple Pivi
of an irreducible representation vi and the v ’s are mutually inequivalent.
I f
P is an irreduc ble sub-representation of ’IT, the above discussion
shows that H; is orthogonal to all but one of the Hvj’s and so H; is
contained in one of the Hvj’s. This proves that every subspace H
Vj
is
uniquely determined, namely, it is the subspace of H", generated by the
spaces of the subrepresentations of ’IT equivalent to vi.
Thus, in the decomposition
EB Pmv;" of ’IT(v;, . . . , v;,.
irreducible and inequivalent), the integers Pi and the class s of the vi are
uniquely determined, just s are the spaces of the Pivi.
References:
2.4.1. PROPOSITION. Let A be an involutive
algebra.
E H, then x ~ (’IT(x)g Ig) is
a positive form on A.
(ii) Let ’IT and ’IT’be representations o f A in Hand H’,
and
f) be a cyclic
vector for ’IT (resp. ’IT’).I f (’IT(x)g Ig) =(’IT’(x)fIf) for every
x E A, there is a unique isomorphism o f H onto H’
mapping
1I’IT(x)gI12 ~ 0,
which gives (i). Now suppose the conditions of (ii) are satisfied. For any
x, yE A we have
(’IT(x)g
!’IT(y)g)
(’IT’(y*x)fIf ) = (’IT’(x)f1’IT’(y)f).
Since the 'IT(x)~ (resp. ’IT’(x)f) are dense in H (resp. H’) it follows that
there is an isomorphism U of H onto H’ such that U( 'IT(x)~) = ’IT’(x)f
for any x E A. We show that U transforms ’IT into ’IT’,i.e. that U’IT(x) =
’IT’(x)U for each x E A;
for every yEA , we have
(U’IT(x»(’IT(y)g)
 
[CR. 2, §4
Since the 7T(Y)~ are dense in H, it follows that U7T(X)
=7T’(X)U.
U7T(X)~) =(U~ 17T’(X)f),
which implies that f = U~. Finally, the uniqueness of U is immediate
since the values that U takes on the (dense) set of t e 7T(A)~ are
predetermined.
2.4.2. In the above notation, the form x ~ (7T(X)~ I~) on A is called the
~
[orms associated with 7T.
S is a set of representations of A, the forms
associated with S are just the forms associated with the various ele›
ments of S.
Let H be a Hilbert space, B an involutive subalgebra of .2(H) and ~
an element of H. We denote by we the positive form on B defined by the
identical representation of B and ~, i.e. the form x ~ (x~ I~). A positive
form on B is said to be a vector form if it is equal to we for a suitable
choice of ~ in H.
2.4.3. Let A be an involutive Banach algebra with an approximate
identity (u;), 7T a non-degenerate representation of A in H, ~ an element
of Hand f the positive form defined by 7T and f Then Iitli= (~ Ig). In
fact, by 2.1.5 v),
=lim f(uj) =lim( 7T(U;)g I g),
and 7T(U;) tends strongly to I (2.2.10). I t follows that if A is the
involutive algebra obtained by adjoining an identity to A, and j and iT
are the canonical extensions of f and 7T to A, then
j(x) = (iT(x)~
Ig) for eac x EA.
In particular, if 7T is the identical representation, assumed to be non›
degenerate, of a sub-C*-algebra A
of 2(H) which does not contain I,
then the canonical extension to A =
A +C . I of we IA is WE IA.
2.4.4. PROPOSITION. Let A be an involutive Banach algebra with an
approximate identity, A the involutive
algebra obtained by
positive fonn on A, j its canonical extension
to A, N the left ideal o A consisting o f
those x E A such that j(x*x)
= 0,
Ai
which is the completion of
 
39
in AIN obtained from left multiplication by x in A by passage to the
quotient. Let g be the canonical image of 1 in A
f
.
(i) Each 7T’(X) has a unique extension to a continuous l inear operator
7T(X) in At•
(iii) g is a cyclic vector for 7T(A).
(iv) f(x) =(7T(X)g Ig) for each x E
A.
(7T’(X )7T’(y)g I7T’(X )7T’(y)g) =/ (y*x*
xy)
7T’(y)g 17T’(y)g),
from which (i) follows. It is plain that 7T’ and then 7T also, are represen›
tations in the sense of the algebra structure. F r x, Y, z E A, we have
(7T(X)7T(y)g !7T(Z)g) =j{z*(xy» =j{(x*z)*y)=(7T(y)g 17T(X*)7T(Z)g),
which implies that 7T(X)* =7T(X*), and (ii) is proved. The set 7T(A)g is the
canonical image of
this proves (iii). Finally, we have for
each x E A,
(7T(X)g g) fO*xl}
vector g
we denote them by
7Tt and gt.
2.4.5. With the above notation, let M be the left ideal of
A
consisting of
those x E A such that f(x*x) =O. The canonical image of A in At can be
identified with
as the completion
x E A, 7T(X)
operator
in
AIM
the
introduction
of
A
and
1. although, in this approach, the definition of g would be less simple.
2.4.6. Let A be an involutive Banach algebra with an approximate
identity, f a continuous positive form on A,
and 7T and g the represen›
tation and
defined by f. Then by 2.4.4 (iv) f is precisely
the
positive form defined by 7T and g.
Conversely, starting with a representation 7T of A in a Hilbert space
and a cyclic vector g for 7T, let f be the positive form defined by 7T and g,
which is continuous as 7T is.
Let 7T’ and f
be the representation and
[CH. 2, §4
(7i(x)g
1
g)
x E A,
and f is a cyclic vector for 7i’. By 2.4.1 (ii), there is a unique isomor›
phism of H", onto H"" mapping 7i to t t’ and g to f.
In particular, let t t be a non-trivial topologically irreducible represen›
tation of A. Every non-zero vector of H", is a cyclic vector for tt , and so
t t is defined up to equivalence by any non-zero form associated with tr.
2.4.7. PROPOSITION. Let A be an involutive
Banach
each x E A,
117i(x)!f = sup f(x*x),
associated with 7i such that
11111::;;; 1.
Thanks to 2.2.6, we need only consider the case of non-degenerate t t ,
By 2. .3, the positive forms associated with 7i of norm ::;;;1 are just the
forms
Thus
117T(X)11
2
lIell")
2.4.8. PROPOSITION. Let A be an involutive Banach algebra with an
approximate identity, f a continuous positive
form on
A and
f.
(i) I f Xo E A, the form x ~ f(x~xxo) is associated with at,
(ii) If f’ is a positive from associated with tt , f’ is the limit (in the norm
topology) of
o
and we have (i).
41
7T
and f’ =wf 0 7T. For every E >0 there is an Xo
E A
such that
117T(Xo)~ - fll
If’(x) - f(x~xxo)1 =1(7T(X)f I
f)
=Ilxll(2Ellfll
+E
and 2Ellfll+E is arbitrarily small.
2.4.9. Let A be a C*-algebra, I a closed two-sided ideal of A, B the
C*-algebra AI I and w: A ~ B the canonical morphism. I f is a positive
form on A such that f(l) =0, then f defines a positive form g on B by
passage to the quotient. For each x E
A, we hav
so that 7T
g
0
W and ~g may be identified with 7Tf and ~f respectively
(2.4.1).
PROPOSITION.
Let A be a C*-algebra, f a positive form on A and I a
closed two-sided ideal of A. Then f(l) =0 if and
only if 7Tf(l) =
so
then
2.4.10. COROLLARY. Ker
contained in Ker f.
2.4.11. COROLLARY.
Let A be a C*-algebra and f and g positive forms
on A. Then the following conditions are equivalent:
(i) Ker 7Tf
C Ker 7T
2.5. Pure forms and irreducible representations
We know how to associate representations with positive forms, and
we now settle the question of when this procedure leads to irreducible
representations.
[CH. 2, §5
2.5.1. PROPOSITION. Let A be an involutive algebra, 7T a representation
of A in H, g an element of
Hand f the positive form defined on A by 7T
and g.
(i) If T is an hermitian operator on H which commutes with 7T(A) and
satisfies 0 ~ T
2g)
on A is a positive form fT which is dominated by f.
(ii) If g is a cyclic vector for 7T, the map T ~ [r is injective.
(iii) If A is an involutive Banach algebra with an approximate identity,
every continuous positive form on A which is dominated by f can be
written fT for some T.
Since fT =CUT{ 0 7T, fT ;;;’0.I f x E A,
fT(x*x) = (7T(x*x)Tg ITg) = 117T(X) Tgl1
2
(7T(X)g
2g
=T,2g if g is a cyclic vector for 7T(A). Since g
is then a separating vector
for the commutant of 7T(A) (A 14) it foll ws
that
T
2
T’ as T;;;. 0, T’;;;. O. This proves (ii).
Let g be a positive form on A which is dominated by f. For x, y A
Ig(y*x)1
•1I7T(y)gW•
on the subspace 7T(A)g of
H which is clearly positive and hermitian. There then exists an her›
mitian operator
0 ~ To ~ I and
= (7T(Z*y)g IT
REPRESENTATIONS 43
invariant under 7T(A) and so P
x
x
is an
hermitian operator in H, lying between 0 and I, which commutes with
the 1T(Z)’S (z E
A). Let T be its positive square root, which again
commutes with the 7T(Z)’S. Then O~ T ~ 1, and
g(y*x)
Finally, suppose
g(y*) =
Let A be a normed involutive algebra and f a
continuous positive form on
A. f is said to be pure if f¥- 0 and if
every
continuous positive form on A which is dominated by f is of the form
>..f(0 ~ A
set of
compact
space and A the C*-algebra of continuous
complex-valued functions on n that vanish at infinity. The pure positive
forms on A may be identified with
the
positive measures on n whose
support consists of a single p int, i.e. with measures of
the
form
f ~
Af(w)
is a fixed point of n: it follows from this
that the pure states of a commutative C*-algebraare just the characters
of the algebra.
2.5.3. Let A be an involutive Banach algebra with an approximate
identity, A the
identity to A,
f a continuous positive form on A and f its canonical
extension to A. Then f is pure if and only if f is pure. Indeed, to begin
with, the conditions
=0 and f =0 are equivalent. Moreover, as g runs
through the set of continuous positive forms on A dominated by
f,
g
runs through the set of continuous positive forms on A dominated by f
(2.1.7). Finally, for g to be of the form Af where 0 ~ A ~ I it is nec ssary
and sufficient
that
g
= At
2.5.4. PROPOSITION. Let A be an involutive Banach algebra with an
approximate identity, f a continuous positive form on A and 1T the
representation of A defined by f Then 1T is non-trivial and topologically
irreducible
 
Let ~ be he
f.
Suppose f is pure, and let E be a projection in H", which commutes
with 7T(A). The
form x ~ (7T(x)E~ IE~) on A is continuous and positive
and is dominated by f [2.5.1 (i)], and is therefore equal to Af for some
o«
By 2.5.1 (ii), E =11.
1
1 and so E =0 or 1. Furthermore, there exist
x E A such that f(x):j; 0
and hence such that (7T(X)~ I~):j; 0,
which
Now suppose 7T is non-trivial
and
topologically irreducible. There
exist x E A such that (7T(X)~ I~) :j; 0 and hence f:j; O. Let g be a con›
tinuous positive form on A dominated by
f.
By
2.5.1
T
7T
p.,
proposition is proved.
P(A)~A.
This map is surjective by 2.4.6 and 2.5.4, and the inverse image in P (A)
of 7T
E A is the set of states associated with 7T (on this subject, d . 2.5.7).
2.5.5. PROPOSITION. Let
A be an
algebra with an
approximate identity and
B the set o f continuous positive forms on A o f
norm ~
I.
(i) B is convex and compact in the weak*-topology (T(A’, A) of the
dual A’ o f A.
(ii) The extreme points o f P consist
o f 0 and the
pure states.
o f
pure states.
B is a weak*-closed convex subset of the unit ball of A’.
This ball is
We
next
show
that
0 is an extreme point of B. I f fE B and -
fEB ,
then f(x*x) = 0 for each x E A, so that If(xW ~ Ilfllf(x*x)
= 0 (2.1.5);
hence f= O.
Now let f be a pure state, and suppose f =Afl +(1 - A)f2 with 0 < 11.<
1, fl ’ f2E B. Then Afl is dominated by f so that Afl =p.,f with 0 ~ p.,
~ 1.
Since
and IlfIII, IIf211:o;:; 1 we must
have Ilf,11
therefore A = I.t and
f. =f = f2’ We have thus proved that f is an extreme point of B.
Conversely, let f be a .non-zero extreme
point of B. Clearly
I,
f2
continuous
positive and non-zero. Put
11f,1I =A, so that IIf211 = 1- A, and let g, =A
"t;
=
=(1 - A)f
from which it follows that f is a pure state . This
proves (ii).
Finally (iii) follows from (i) .while (ii) follows from the
Krein-Milman
theorem.
unital. B is then
such
the set
E(A) of states of A is the set of positive forms f on A such
that
f(1)
se t of extreme
set of those extreme points of B
which belong to E(A), i.e. the set P (A).
E(A)
7T
~2 two vectors of H",
and fI and f2 the positive forms defined by (7T, ~,) and
(7T, ~2)' Then f, = f2
if and only if there is a complex number
A of absolute value
are cyclic vectors for 7T (2.3.1), there is an
automorphism U of H commuting with the 7T(X)’S such that
U~I = ~2
(2.4.1 (iij). Now
IAI =1.
In particular, the canonical map P (A) ~ A is bijective if and
only if
topologically irreducible representation of A is one-dimensional.
When A is a C*-algebra, theorem 2.7.3 implies that this condition is
fulfill d if and
References: [618], [619], [638
2.6.1. THEOREM. Any
a Hilbert space.
be the real Banach space consisting of the hermitian elements
of A, and let x be a non-zero element of A. By (1.6.1), - x*xe A+, and
since A + is a closed convex cone (1.6.1), there is a continuous linear
form
t,
+ and
Identifying
fx
with an hermitian form on A, we see that t, is a positive
form on A and fx(x*x) >0, so that the representation 7T
x
# O. Let the representation 7T be the direct
sum of the 7Tx’S for x E A, x:;t: O.
Then 7T is injective and therefore
isometric (1.3.7 and 1.8.1) and the theorem is proved.
Thus, as we previously asserted, the closed involutive sub-algebras of
Ie(H)
for H a Hilbert space are indeed the most general examples of
C*-algebras.
2.6.2. PROPOSITION. Let A be a C*-algebra and let x E A. Then the
following conditions are equivalent:
O.
(ii) The operator 7T(X) is ;;;;. 0 for every representation 7T of
A.
(iii) f(x);;;;. 0 for every positive form f on A.
(i) ~ (iii): obvious.
A
the form
;;;;.
0 if (iii) holds. The
operator 7T(X) is thus > O.
(ii) ~ (i): let 7T be an isometric representation of A (2.6.1); if the
operator 7T(X) is ;;;;.0, then
7T(X) is positive in 7T(A) (1.6.5) and so x is
positive in A.
h
the
set
of hermitian
elements of A, B the set of positive forms on A of norm ~ 1 and PCB
the set of pure states
of
A,
weak*-topology. Let cg(B), cg(P) be the sets of
continuous real-valued
functions on
h
let F, be the
continuous real-valued function f -+ <I, x) on B and Ox its restr ction to P.
Then the map x ~ F, (resp. x -+ Ox) is an isometric isomorphism
of the
ordered Banach
space A
cg(B) [resp. cg(P)].
x ;;;;.0 ~ F, ;;;;.0 ~
Ox ;;;;. O.
 
x ~ 0 (2.6.2). Now suppose A
is
H (2.6.1) and let x E A
h
I(xg
I
g)1
there exist fEB
arbitrarily close to 1 and
therefore also f E P
with If(x)! arbitrarily close to 1. Consequently, 1 ~ IIOJ ~ Fxll ~ 1 and
the
corollary follows.
When A is commutative, the map x ~ Ox is simply the Gelfand
isomorphism restricted to A
fand isomorphism to the
some ways, will be studied later on (10.5.4).
2.6.4. COROLLARY. Let A be a C*-algebra and g a continuous hermitian
linear f rm on A. Then there exist positive forms f, I’ on A such that
g =f - 1’, Ilgll= 11111
+Ilf’ll•
A
h
p. on Cf5(B)
can be written
p. =p.+- P.- where p. +, P.- are positive measur s such that lip. II =
lip. +11
h
so that
Ilgll= 11111
11f’11•
We shall see later (12.3.4) that the decomposition of 2.6.4 is unique.
References: [618], [619], [681], [848], [1097], [1101], [1323], [1455],
[1589].
2.7. The enveloping C*•algebra of an involutive Banach algebra
2.7.1. PROPOSITION. Let A be an involutive Banach algebra with an
approximate identity. Let R be the set
of
of
on A, and P the set of pure states of
A. For
(1) sup 117T(x)1I =
1TER 1TER’ fEB fEP
of
II’, we
[CH. 2, §7
have Ilxll’",;;;[x]. The map x ~ Ilxll’ is a seminorm on A such that
Ilxyll’",;;;llxll’llyll’, Ilx*II’=llxll’,
Ilx*x]’
for any x, yEA.
We denote the four numbers in the order in which they appear in (1)
by a, b, c, d.
d e: b: let f E
P ; then f is associated with a representation 1T E R’
(2.5.4) and f(x*x)",;;; 111T(X)W (2.4.7);
b ",;;; a: obvious;
c: this follows from 2.4.7;
c"’;;;d: let fEB; by 2.5.5, f(x*x) lies in the c losed interval whose
left-hand end-point is 0 and whose right-hand
end-point is SUPgEP
f(x*x)"’;;;SUPgEPg(X*X).
By 1.3.7, 111T(X)II",;;; Ilxll for every 1T E R, so that IIxll’",;;;[r]. Moreover,
each function x ~ 1I1T(x)1I is a seminorm on A, hence x ~ IIxll’ is a
seminorm on A. We have
for each 1T E R, and
so
111T(XY)II",;;; 111T(x)II•I/1T(y)ll",;;; Ilxll’ •llyll’, hence
Ilxyll’",;;;
Ilxll’ •llyll’•
2.7.2. Let I be the set of x E A such that Ilxll’ = 0, which is a closed
self-adjoint
x ~ IIxll’
the C*-algebra
general. The completion B of
AI I is a C*-algebra
called the enveloping C*-algebra of A. The canonical
map of
is dense in B.
2.7.3. When A is a IIxl/’ = IIxll by 2.6.1, and A may be
identified with its enveloping C*-algebra.
In
re›
claimed, and considerably strengthens
49
THEOREM. For any C*-algebra A, there is a family (7T;) of topologi›
cally irreducible representations
every x E A.
2.7.4. PROPOSITION. Let A be an involutive Banach algebra with an
approximate identity, B the enveloping C*-algebra of
A and T the
(i) I f
7T
is a representation of A, there is exactly one representation p
of B such that 7T = pOT, and
pCB) is the C*-algebra generated by 7T(A).
(ii) The map 7T ~ P is
a bijection of the set of representations o f A
onto
the set of representations of B.
(iii) p is non-degenerate (resp. topologically irreducible) if and only if
7T is non-degenerate (resp. topologically irreducible).
Let
7T be a representation of A. In th notation of 2.7.2,
7T vanishes on
I, and defines, by passing to the quotient, a representation 7T’ of All
such that 117T’(z)11 ~ Ilzll for each
zEAl I,
norm
of
B.
7T’
therefore extends to a representation p of B which satisfies 7T = pOT.
The uniqueness statement of (i) follows from the fact that
T(A)
is dense
in B. This also implies that 7T(A) is dense in pCB) in the operator-norm
topology. Since pCB) is a C*-algebra (1.8.3), it is the C*-algebra
generated by 7T(A). Statements (ii) and (iii) are immediate.
Proposition 2.7.4 shows that B is the so ution of a universal problem.
It
also shows that in the majority of questions concerning represen›
tations of involutive Banach algebras with an approximate identity, it is
enough to deal only with the C*-algebra case.
2.7.5. PROPOSITION. With A, B, T as in 2.7.4:
(i) If f is a continuous positive [orm on A, there is exactly one
positive form g on B such
that f = gOT. Moreover Ilgll
= 11!l1.
(ii) The map f ~ g is a bijection of the set of
continuous positive forms
on A onto the set of positive forms on B.
(iii)
Let
M be a bounded set of continuous positve forms on A. Then
the map f ~ g, restricted to M, is bicontinuous for t e weak*-topologies
u(A’,
A), at B’,B), where ’, B’ denote the duals of A and B respec›
tively.
Let f be a continuous positive form on A. Then for each x E A we
have, using 2.1.5 (i),
g
gO and
n
n
g(y*y) =lim f(x~xn) ~ 0,
and so g is positive. I f x E A and Ilxll ~ 1, then
If(x)1
and hence 1If11 ~ "gil•
The uniqueness of g follows from the fact that T(A) is dense in B. We
have now proved (i), and (ii) is immediate. Let MeA ’ be a set of
continuous positive forms on A and NcB ’ its image under the map
f ~ g. The map f ~ g of M onto N is plainly bicontinuous for the
weak*-topologies £T(A’, A), £T(B’, T(A». I f M is bounded, N is bounded
by (i), so that
B.
2.7.6. Let A be an involutive Banach algebra with an approximate
identity, and let Q be the set of continuous positive forms on A. For
each f
7T = E9
called the universal representation of A, and is non-degenerate. By
(2.2.7) and (2.4.6), every non-degenerate representation of A is a direct
sum of representations of the form 7Tf’ so that IIxll’=
117T(x)1I
for every
x E A. Hence i B is the enveloping C*-algebraof A, the representation
of B corresponding o 7T is an isomorphism of B onto 7T(A).
References: [582], [619], [638], [1097], [1101], [1323], [1455].
2.8. A theorem on transitivity
2.8.1. LEMMA. Let H be a Hilbert space, (gl,
. . . , gn) an orthonormal
’I",
.,’n vectors in H of norm ~ r. Then there is a
b E .P(H) of
’I,... hg
Let K be the subspace of H generated by
g., ... ,
b as an operator which leaves K invariant and which
vanishes on He K. This reduces us to the case where K = H. Now let
(gl,
. . . , gn, gn+(, . . . , gm) be an orthonormal basis for H, and let b be the
 
51
(a) the first n columns are the coordinates of ’I’... ,’n with respect to
the ~i;
(b) if h exists, the first n rows are the complex conjugates of the first
n columns (th s is possible since the existence of h implies
that
the
n
We have
z
2.8.2. LEMMA. Le HI, . . . , H
p
strongly dense
thonormal system in H and Tit, , TJn be vectors
in H of norm ,,;;; r,
where
1,
n, ~i and TJi both lie in the same subspace
Hj(i). Then there exists abE A o norm ,,;;;3m I/Z such that b ~ 1 =
TJl’ . . .
h of .2(H) such that
h~ 1 = TJI’ , h~n = TJ.. then b can in addition be chosen to be an
hermitian element of A.
By Kaplansky’s density theorem, the unit ball of A (respectively of
the hermitian part of A) is strongly dense in the unit ball of .2(H
I)
x
1
»
Now by 2.8.1 applied to the spaces HI>"" H; in turn, there is a
Yo E .2(H
) such that
and then by the preceding remark there is Xo E A such that
Next, again by 2.8.1, there exists
Yl
r
and then XI E A such that
 
.2(H]) x . . . x .2(H
(If
h exists, the Yk and Xk can be chosen to be hermitian.) x
o
+XI +.. .
then converges in norm to an element b E A such
that
bg]=
’TIl’
•)(2n)I/2r :s;;; 3m 1/2.
2.8.3. THEOREM. Let A be a C*-algebra, A the C*-algebra obtained by
adjoining an identity to A, 7T.,
, 7T
p
spaces H., . . . , H
the 7Tj are
inequivalent.
sional subspaces of HI>
that
7Tj(x)IK
j
= T
j
IK
j
p
7Tj(X)
(iii) Let T,
E .2(H,), . . . , T
C*-algebra of .2(H) which commutes with the projections
E
j
= PHi’
Neumann
7T(A) C .2(H
p
) .
 
strongly dense in B. Since ’TTj is topologically irreducible, E
j
B is ;t(H
j
let j
and k be distinct indi es. I f Fj and F
k
non-zero projections Ej, E
j
j
E;
E
k
=E
b
B’
j
and E, as initial and final projections respectively; ut then ’TTj
and ’TTk are equivalent which is a contradiction.
Hence
the
j
j
I
p
By 2.8.2, there is an x E A such that
’ITj(x) IK, =T
I f
chosen so that
’TT(x)
’TT(!(x+x*», x can
itself be chosen to be hermitian. Now suppose the T/s are unitary. For
j =1, . . .
Kj:2 K, of
j
t ; Itc, =t, IK
Tj
By the foregoing
work, there is an hermitian element y of A such that ’TT(y) IKj = T’IIKj for
each
j.
Then x =exp(iy) is a unitary element of Aand iTj(x) IK, =T, IK, for
each j.
2.8.4. COROLLARY.
C*-algebra is algebraically irreducible.
We have merely to apply theorem 2.8.3 with p = 1 and dim K
I
1.
Thus we will hen eforth speak of irreducible representations of a
C*-algebra without further qualification.
2.8.5. COROLLARY. Let A be a C*-algebra, f a pure positive form on A
and N the left ideal
of
=O. Then AIN, with
is
Indeed, let ’TT be this representation, topologically irreducible by
(2.5.4). From the construction of
’TT, AIN
1T
invariant
under
’TT.
1T
(2.8.4).
[ H. 2, §9
2.8.6. COROLLARY. Let A be a C*-algebra, A the C*-algebra obtained
by adjoining an identity to A, and
t,
and f2 define equivalent representations ’7TI and ’7T2
if and only if there is a
unitary element u of A such that f2(x) = iJ(u*xu) for each x E A.
Let
~ I be the vector of H", defined by fl’ The states of
A that define
representations equivalent to
(2.4.6), i.e. the forms x ~ (l1ix)~ I~) where
~
is a unit vector of H"’I
(2.4.3). Now the unit vectors of H"’Iare just the vectors iil(u)~1 where u
is unitary in A (2.8.3) and ii i is the canonical extension of ’7TI to A.
Finally,
=
2.9. Ideals in C*-algebras
2.9.1. PROPOSITION. Let A be a C*-algebra, f a positive form on A, M
its kernel and N the set of x E A such that f(x*x) =O.
Then
M ~
N +
N*.
=M and so N +
N* ~ M. Now suppose
is pure, and let A be the C*-algebra obtained
by adjoining an i entity to A, and! the canon cal extension of f to A›
Let ’7T and e the irreducible representation and vector defined by f.
Let b EM and let 11 be its canonical image in H",. By the construction of
’7T and ~, (11 I~) =!(1*
b) =0,
and hence there is an hermitian operator
in H", which maps { to zero and leaves 1’/ fixed. By 2.8.3, there is an
hermitian element a of A such that ’7T(a){=0 and ’7T(a)1’/=11, i.e, such
that a
and so M=M*~N*+N.
2.9.2. LEMMA. Let A be a unital C*-algebra, L a closed left ideal of A
and x E A". If, for every E > 0, there is a positive element u. of L such
that x :os::
I f t =
 
Now
II
x
I/
2
=
xEL .
2.9.3. LEMMA. Let A be a C*-algebra and L a closed left ideal of A.
Then A is the closed left ideal generated by L n A+.
This follows immediately from 1.7.3.
2.9.4. LEMMA. Let
A be a C*-algebra and L, L’ two closed left ideals of
A such that L C L’.
Suppose every positive form on A that vanishes on L
also vanishes on L’. Then L =L’.
We may assume A to be unital. Let
a E L’n
+ and let e be >0. The
set S. of positive forms f on A such that f(l) =
!lfli=1
and f(a) ~ e is
weak*-compact. I f f E S., f is not identically zero on
L’
and therefore
nor on L, so that there exists an xf E L such that
f(xf)
¢ 0; consequently
there is a weak*-neighbourhood U, of f in . such that g(xf) ¢ 0 if
g E U
f
, and, by the compactness of S., there are a finite open covering
(U;),.. j<;;m of S. and elements a
l
0<lf(a;)12~f(aia;)
for fE U;.
Hence f(atal + . . . + a:a
m
a:a
m
+ E -
a) ~
that a ~ atal+ . . . +
+ e (2.6.2).
Hence a E L (2.9.2). Land L’ thus contain the same positive elements
and are therefore equal (2.9.3).
&nb