Extending Pure States on C*-Algebras and Feichtinger’s Conjecture Special Program on Operator Algebras 5 th Asian Mathematical Conference Putra World Trade Centre, Kuala Lumpur MALAYSIA June 2009 Wayne Lawton Department of Mathematics National University of Singapore

Basic Notation denote the natural, integer, rational, real, complex numbers. circle group Haar measure

Riesz Pairs satisfying any of the following equivalent conditions Problem: characterize Riesz pairs

Synthesis Operator defines a Synthesis OperatorA subset and the Hermitian form is linear inand congugate-linear in denotes a complex Hilbert space

Bessel Sets admits an extension is a Bessel Set if Then its adjoint, the Analysis Operator, Frame Operator satisfies where exists and and the

Frames that satisfies any of the following equivalent conditions: Proof of Equivalence: [Chr03], pages is a Frame for Example. is surjective, is injective, 3. but not a frame for if it is a Bessel set is a Bessel set, Proof: [Chr03],

Riesz Sets is a Riesz Set if it is a Bessel set that satisfies but never a Riesz set. Example: Union of n > 1 Riesz bases for is always a frame for any of the following equivalent conditions: 1.is bijective, 2. Proof of Equivalence: [Chr03], 66-68, Remark:is the grammian, and dual- grammian used by Amos Ron and Zuowei Shen is bijective, 3. is the

Stationary Sets is a Stationary Set if there exists there exists a positive Borel measure Then the function is positive definite so by a theorem of Bochner [Boc57] such that on such that and a unitary Example

Stationary Sets is stationary set thenIfis a 1. Bessel set iff there exists a symbol function is a 4. Riesz set iff 2. Frame iff Proof. [Chr], and then 3. Tight Frame iffis constant on its support

Stationary Bessel Sets with symbol Representation as Exponentials Representation as Translates

- Riesz set is one satisfying - Conjecture: For everyevery Riesz set is a finite union of- Riesz sets Feichtinger Conjecture: Every Bessel set is Feichtinger set Definition LetAn Two Conjectures Definition A Fechtinger set is a finite union of Riesz sets

Pave-able Operators is pave-able if and a partition whereis the diagonal projection (1) Theorem 1.2 in [BT87] There exists density such that with positive satisfies (1) Observation This holds iff for everythe columns of are a finite union of-Riesz sets

States on C*-Algebras Examples - algebra that satisfies any of the following equiv. cond. is a linear functionalA State on a unital is convex and weakly compact Krein-Milman  A 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 Remarks on have a unique extension to a stateon Hahn-Banach  extensions always exist Problem arose from Dirac quantization conjectures - Paving Conjecture: every - other conjectures in mathematics and engineering is pave-able

Let conjectures. then [HKW86,86] If Two Conjectures for Stationary Sets is Riemann integrable then be a Bessel set with symbol satisfies both the Feichtinger and Theorem 4.1 in [BT91] If 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 satisfies Feichtinger’d conjecture iff is a Riesz pair where

with symbol Feichtinger Conjecture for Stationary Sets where the closure is wrt the hermitian product Corollary Never for is a Riesz set. then we call We consider a stationary Bessel set Then whereis a fat Cantor set Definition If a Riesz pair if Theorem If then is a Riesz pair iff

New Results Theorem 1. If Pseudomeasure this happens if then ‘contains’ a point measure

where New Results is a compact Corollary 1. If Remark Characteristic functions of Kronecker sets are we call and Definition Given a triplet is a homomorphism, is a fat Cantor setis a Kronecker set and topological group, is an open neighborhood of the the identity in a Kronecker set. uniformly recurrent points in the Bebutov system [Beb40] thenis 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 and a thick set [F81]. is syndetic if there exists Theorem 1.23 in [F81] page 34. If is a partition then one of theis piecewise syndetic. Observation in proof of Theorem 1.24 in [F81] page 35. If is piecewise syndetic then the orbit closure of contains the characteristic function of a syndetic set. Theorem 2.satisfies Feichtinger’s conjecture iff is a Riesz pair for some syndetic (almost per.)

References S. Bochner, Lectures on Fourier Integrals, Princeton University Press, 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), 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), M. Bownik and D. Speegle, The Feichtinger conjecture for wavelet frames, Gabor frames, and frames of translates, Canad. J. Math. 58#6 (2006), H. Bohr, Zur Theorie der fastperiodischen Funktionen I,II,III. Acta Math. 45(1925),29-127;46(1925), ;47(1926), 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 P. G. Casazza, O. Christenson, A. Lindner, and R. Vershynin, Frames and the Feichtinger conjecture, Proc. Amer. Math. Soc. 133#4 (2005), 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 P. G. Casazza and E. Weber, The Kadison-Singer problem and the uncertainty principle, Proc. Amer. Math. Soc. 136 (2008),

References O. Christenson, An Introduction to Frames and Riesz Bases, Birkhauser, H. Halpern, V. Kaftal, and G. Weiss, Matrix pavings and Laurent operators, J. Operator Theory 16#2(1986), R. Kadison and I. Singer, Extensions of pure states, American J. Math. 81(1959), H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton Univ. Press, W. H. Gottschalk and G. A. Hedlund,Topological Dynamics, Amer. Math. Soc., Providence, R. I., N. Weaver, The Kadison-Singer problem in discrepancy theory, preprint H. Halpern, V. Kaftal, and G. Weiss,The relative Dixmier property in discrete crossed products, J. Funct. Anal. 69 (1986),