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 Singaporematwml@nus.edu.sg

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.

N. Weaver, The Kadison-Singer problem in discrepancy theory, preprint

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