Extending Pure States on C*-Algebras and Feichtinger’s Conjecture
Special Program on Operator Algebras5th Asian Mathematical Conference
Putra World Trade Centre, Kuala LumpurMALAYSIA 22-26 June 2009
Wayne Lawton
Department of Mathematics
National University of [email protected]
http://www.math.nus.edu.sg/~matwml
Basic NotationCRQZN
denote the natural, integer, rational, real, complex numbers.
circle groupZR /||),/( SZRBorelS Haar measure
ZR
ghghZRLgh/
2 ,),/(,
ZR
ghghZRL/
,),/(
Riesz Pairs
ZZZRBorelSZS 11 ),/(),,(satisfying any of the following equivalent conditions
)/(0.1 2 ZRLh
S
hZhh 21 ||ˆsup1,||||
IQP 11 0.2
gQggPg ZZS ˆ)(,1\
^
Problem: characterize Riesz pairs
Synthesis Operator
HB defines a Synthesis Operator
),,(H
A subset
HBCT )(:finfin
BbbbffT )(fin
and the Hermitian form ghghHH ,),(
is linear in h and congugate-linear in .g
denotes a complex Hilbert space
Bessel Sets
admits an extension
is a Bessel Set if
HBT )(: 2
Then its adjoint, the Analysis Operator,
)(: 2 BHT
bhbhT ,))((
B
Frame Operator HHTTS : satisfies
IS where2|||||||| TS
exists and
HBCT )(:finfin
and the
Frames
that satisfies any of the following equivalent conditions:
B
1.
GG SI |0 2.
Proof of Equivalence: [Chr03], pages 102-103.
is a Frame for
GBT )(: 2
Example.
is surjective,
)(:| 2 BGT G is injective,
3.
},{ 1 NkeeB kk
).(span)(2 BNH but not a frame for
if it is a Bessel set
is a Bessel set,
Proof: [Chr03], 98-99.
HBG )(span
Riesz Setsis a Riesz Set if it is a Bessel set that satisfiesB
but never a Riesz set.
Example: Union of n > 1 Riesz bases for
is always a frame for H
any of the following equivalent conditions:
1. )(span)(: 2 BGBT is bijective,
2.
Proof of Equivalence: [Chr03], 66-68, 123-125.
,H
TT Remark: is the grammian, and
dual- grammian used by Amos Ron and Zuowei Shen
TTS
http://www.math.nus.edu.sg/~matzuows/publist.html
)(:| 2 BGT G is bijective,
3. TTIB
)(20
is the
Stationary Setsis a Stationary Set if there existsB
HHU :
there exists a positive Borel measure
Then the function
is positive definite so by a theorem of Bochner [Boc57]
hUhkgCZg k,)(,:
Hhsuch that
dv on
such that
and a unitary
}.:{ ZkhUB k
ZR /
.)(/
2 ZR
dvekg ix
Example
ikxk exhUhdvZRLH 22 ))((,1),,/(
Stationary Sets
is stationary set thenIf B is a
1. Bessel set iff there exists a symbol function
dxxdvZRL )()/(
B is a
4. Riesz set iff
)(0)(,/0 xxZRx 2. Frame iff
)(,/0 xZRx
Proof. [Chr], 143-145.
and then
B
3. Tight Frame iff is constant on its support
Stationary Bessel Setswith symbol
))(,/(2 dxxZRLH Representation as Exponentials
Representation as Translates
}:)({ ZkkmhB
}:{ 2 ZkeB ikx
)/( ZRL
gghZH ,ˆ),(2
)(ˆ),( 22 mkee ikximx
)(ˆ))(),(( mkkhmh
- Riesz set is one satisfying
ITTI )1()1(
- Conjecture: For everyR
,0 every Riesz set is a
finite union of - Riesz sets
Feichtinger Conjecture: Every Bessel set is Feichtinger set
Definition Let .0 An
Two ConjecturesDefinition A Fechtinger set is a finite union of Riesz sets
Pave-able Operators
is pave-able if Nn ,0and a partition
||)(||||))((|| bdiagbPbdiagbPjj
))(( 2 ZBb nZZZ 1
where )()(: 22 ZZPj is the diagonal projection
jZk kkZk kkj ececP
(1)
Theorem 1.2 in [BT87] There exists ZZ 1density such that
with positive
satisfies (1)1P
Observation This holds iff for every
bthe columns of0
are a finite union of -Riesz sets
States on C*-AlgebrasA
vavaZvZB ,)(),(,)( 22
Examples
- algebra
that satisfies any of the following equiv. cond.
)()(,),()( paaZpZCZ
CA:is a linear functional CA State on a unital
1)( and 1|||| I1.
2. 0)(A,a and 1|||| aa} on states{ A is convex and weakly compact
Krein-Milman }states pure{convA Pure State is an extremal state
The Kadison-Singer Problem
Does every pure state
YES answer to KS is equivalent to:
- combination of the Feichtinger and
ppA eaeaZZp ,)(|~~ !
)(ZA
Remarks
on
have a unique extension to a state ~ on ?))(( 2 ZB
Hahn-Banach extensions always exist
Problem arose from Dirac quantization
R conjectures
- Paving Conjecture: every
- other conjectures in mathematics and engineering
))(( 2 ZBb is pave-able
Let
conjectures.R
then
[HKW86,86] If
BTwo Conjectures for Stationary Sets
is Riemann integrable then
be a Bessel set with symbol )/( ZRL
Bsatisfiesboth the Feichtinger and
Theorem 4.1 in [BT91] If |||)(ˆ|0
2kk
Zk
B,0 is a finite union of
Corollary 4.2 in [BT91] There exist dense open subsets of
R/Z whose complements have positive measure and whose characteristic functions satisfy the hypothesis above.
Observation The characteristic functions of their complementary ‘fat Cantor sets’ satisfy both conjectures
-Riesz bases
Observation B satisfies Feichtinger’d conjecture iff 0),(,1 in ZSZZZ is a Riesz pair where
})(:/{ xZRxS
1
0)(0
n
j njx
with symbolB
Feichtinger Conjecture for Stationary Sets
where the closure is wrt the hermitian product
nZmZNnZm 1,,
Corollary Never for
is a Riesz set.
ZR
ghgh/
,
then we call
C
We consider a stationary Bessel set
Then )(span:2 BHZkeB ikx
where C is a fat Cantor set
Definition If ZZ 1),( 1Z a Riesz pair if
12
1 : ZkeB ikx
Theorem If then ),( 1Zis a Riesz pair iff
New Results
)(ˆ)/( ZLZRM
Theorem 1. If
YhhZRLh )sup(,1||||),/(.1 2
Pseudomeasure
ZZZRBorelS 1),/(
this happens if
then
1)sup(),/(, ZvZRM ZRSY /)sup(
S
fhf 22 ||ˆ||||)(.2
.4
not RB ),(0||||/||inf.3 122 ZSff
S
‘contains’ a point measure
where
)1())((,}1,0{),,( ksksZ
New Results
is a compact),,( VX
Corollary 1. If
Remark Characteristic functions of Kronecker sets are
we call
and
1Z
Definition Given a triplet
is a homomorphism,
is a fat Cantor setSis a Kronecker set and
Xtopological group, XZ :
V is an open neighborhood of the the identity in ,X)(),,( 1 VVXZ a Kronecker set.
uniformly recurrent points in the Bebutov system [Beb40]
then ),( 1ZS is not a Riesz pair.
This notion coincides with almost periodic in [GH55].
New Results
and piecewise syndetic if it is the intersection of a syndetic
thick if
Definitions A subset
nZZZ 1
and a thick set [F81].
ZZ 1 is syndetic if there exists
},1,...,2,1,0{1 nZZNn,}1,...,2,1,0{ 1ZnmZmNn
Theorem 1.23 in [F81] page 34. If
is a partition then one of the iZ is piecewise syndetic.
Observation in proof of Theorem 1.24 in [F81] page 35. If iZis piecewise syndetic then the orbit closure of
iZ
contains the characteristic function of a syndetic set.
Theorem 2. B satisfies Feichtinger’s conjecture iff ),( 1Zis a Riesz pair for some syndetic (almost per.) 1Z
References
S. Bochner, Lectures on Fourier Integrals, Princeton University Press, 1959.
J. Bourgain and L. Tzafriri, On a problem of Kadison and Singer, J. Reine Angew. Math. 420(1991), 1-43
J. Anderson, Extreme points in sets of positive linear maps on B(H), J. Func. Anal. 31(1979), 195-217.
J. Bourgain and L. Tzafriri, Invertibility of “large” submatrices with applications to the geometry of Banach spaces and harmonic analysis, Israel J. Math., 57#2(1987), 137-224.
M. Bownik and D. Speegle, The Feichtinger conjecture for wavelet frames, Gabor frames, and frames of translates, Canad. J. Math. 58#6 (2006), 1121-2243.
H. Bohr, Zur Theorie der fastperiodischen Funktionen I,II,III. Acta Math. 45(1925),29-127;46(1925),101-214;47(1926),237-281
M. V. Bebutov, On dynamical systems in the space of continuous functions, Bull. Mos. Gos. Univ. Mat. 2 (1940).
References
P. G. Casazza and R. Vershynin, Kadison-Singer meets Bourgain-Tzafriri, preprint
www.math.ucdavis.edu/~vershynin/papers/kadison-singer.pdf
P. G. Casazza, O. Christenson, A. Lindner, and R. Vershynin, Frames and the Feichtinger conjecture, Proc. Amer. Math. Soc. 133#4 (2005), 1025-1033.
P. G. Casazza, M . Fickus, J. Tremain, and E. Weber, The Kadison-Singer problem in mathematics and engineering, Contep. Mat., vol. 414, Amer. Math. Soc., Providence, RI, 2006, pp. 299-355.
P. G. Casazza and E. Weber, The Kadison-Singer problem and the uncertainty principle, Proc. Amer. Math. Soc. 136 (2008), 4235-4243.
ReferencesO. Christenson, An Introduction to Frames and Riesz Bases, Birkhauser, 2003.
H. Halpern, V. Kaftal, and G. Weiss, Matrix pavings and Laurent operators, J. Operator Theory 16#2(1986), 355-374.
R. Kadison and I. Singer, Extensions of pure states, American J. Math. 81(1959), 383-400.
H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton Univ. Press, 1981.
W. H. Gottschalk and G. A. Hedlund,Topological Dynamics, Amer. Math. Soc., Providence, R. I., 1955.
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