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Extending Pure States on C*-Algebras and Feichtinger’s Conjecture Special Program on Operator Algebras 5th Asian Mathematical Conference Putra World Trade Centre, Kuala Lumpur MALAYSIA 22-26 June 2009 Wayne Lawton Department of Mathematics National University of Singapore [email protected] http://www.math.nus.edu.sg/~matwml Basic Notation N Z Q RC denote the natural, integer, rational, real, complex numbers. R / Z circle group S Borel ( R / Z ), | S | Haar measure 2 h, g L ( R / Z ), h, g h g R/Z L ( R / Z ), h, g h g R/Z Riesz Pairs (S , Z1 ), S Borel ( R / Z ), Z1 Z satisfying any of the following equivalent conditions 1. 0 h L2 ( R / Z ) || h || 1, sup hˆ Z1 | h |2 2. 1 0 P Q 1I Pg S g , S (Qg ) ^ Z \ Z1 gˆ Problem: characterize Riesz pairs Synthesis Operator (H , , ) denotes a complex Hilbert space and the Hermitian form is linear in A subset h H H (h, g ) h , g and congugate-linear in g. B H defines a Synthesis Operator Tfin : Cfin ( B) H Tfin f bB f (b)b Bessel Sets B is a Bessel Set if Tfin : Cfin ( B) H admits an extension T : ( B) H 2 Then its adjoint, the Analysis Operator, T : H ( B) 2 (T h)(b) h, b and the Frame Operator S TT : H H satisfies exists and SI where || S || || T || 2 Frames B is a Frame for G span ( B) H if it is a Bessel set that satisfies any of the following equivalent conditions: 1. 2. 3. T : ( B) G is surjective, 2 T |G : G ( B) is injective, 2 0 I G S |G Proof of Equivalence: [Chr03], pages 102-103. Example. B {ek ek 1 , k N } is a Bessel set, but not a frame for H ( N ) span ( B). Proof: [Chr03], 98-99. 2 Riesz Sets B is a Riesz Set if it is a Bessel set that satisfies any of the following equivalent conditions: T : ( B) G span ( B) 2 2. T |G : G ( B ) is bijective, 1. 3. 2 is bijective, 0 I 2 ( B ) T T Proof of Equivalence: [Chr03], 66-68, 123-125. Example: Union of n > 1 Riesz bases for H H, is always a frame for but never a Riesz set. Remark: T T is the grammian, and S TT is the dual- grammian used by Amos Ron and Zuowei Shen http://www.math.nus.edu.sg/~matzuows/publist.html Stationary Sets B is a Stationary Set if there exists h H and a unitary k U : H H such that B {U h : k Z }. Then the function g : Z C , g (k ) h,U h k is positive definite so by a theorem of Bochner [Boc57] there exists a positive Borel measure g (k ) e such that 2ix dv on R/Z dv. R/Z Example H L ( R / Z , dv), h 1, (U h)( x) e 2 k 2ikx Stationary Sets If B is stationary set then B is a 1. Bessel set iff there exists a symbol function L ( R / Z ) dv ( x)dx and then B 2. Frame iff is a 0 x R / Z , ( x ) 0 ( x ) 3. Tight Frame iff 4. Riesz set iff is constant on its support 0 x R / Z , ( x ) Proof. [Chr], 143-145. Stationary Bessel Sets with symbol L ( R / Z ) Representation as Exponentials H L ( R / Z , ( x )dx ) 2 B {e (e 2imx ,e 2ikx 2ikx : k Z } ) ˆ (k m) Representation as Translates H ( Z ), h gˆ , g 2 B {h(m k ) : k Z } (h( m), h( k )) ˆ (k m) Two Conjectures Definition A Fechtinger set is a finite union of Riesz sets Feichtinger Conjecture: Every Bessel set is Feichtinger set Definition Let 0. An - Riesz set is one satisfying (1 ) I T T (1 ) I R- Conjecture: For every 0, every Riesz set is a finite union of - Riesz sets Pave-able Operators b B( ( Z )) is pave-able if 0, n N and a partition Z Z Z 1 n 2 (1) || Pj (b diag (b)) Pj || || b diag (b) || where Pj : ( Z ) ( Z ) 2 Pj 2 is the diagonal projection c e c e k k k k kZ kZ j Observation This holds iff for every are a finite union of -Riesz sets b Theorem 1.2 in [BT87] There exists density such that satisfies (1) P1 0 the columns of Z1 Z with positive States on C*-Algebras A State on a unital C - algebra A is a linear functional : A C that satisfies any of the following equiv. cond. 1. || || 1 and ( I ) 1 2. || || 1 and a A, ( a a ) 0 {states on A} is convex and weakly compact A Pure State is an extremal state Krein-Milman Examples conv {pure states } ( Z ) C ( Z ), p Z , (a ) a( p) B ( Z ), v ( Z ), (a) av, v 2 2 The Kadison-Singer Problem A (Z ) ~ on B(2 ( Z )) ? have a unique extension to a state Does every pure state on Remarks Problem arose from Dirac quantization Hahn-Banach extensions always exist ~ ~ p Z Z ! |A (a ) ae p , e p YES answer to KS is equivalent to: - combination of the Feichtinger and R conjectures 2 - Paving Conjecture: every b B( ( Z )) is pave-able - other conjectures in mathematics and engineering Two Conjectures for Stationary Sets B be a Bessel set with symbol L ( R / Z ) Observation B satisfies Feichtinger’d conjecture iff 0 Let Z Z1 Z n , ( S , Zi ) is a Riesz pair where S {x R / Z : ( x ) } [HKW86,86] If is Riemann integrable then B satisfies both the Feichtinger and conjectures. R Theorem 4.1 in [BT91] If then 0 kZ | ˆ (k ) | | k | 2 0, B is a finite union of -Riesz bases 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 Feichtinger Conjecture for Stationary Sets We consider a stationary Bessel set B with symbol Then B e 2ikx :k Z H span ( B) where the closure is wrt the hermitian product h, g Definition If Z1 Z h g R/Z then we call B1 e2ikx : k Z1 Theorem If ( , Z1 ) a Riesz pair if is a Riesz set. m Z , n N , Z1 m nZ then ( , Z1 ) is a Riesz pair iff Corollary Never for 0 j 0 ( x ) n 1 C where C j n is a fat Cantor set New Results M ( R / Z ) ˆ L ( Z ) Pseudomeasure S Borel ( R / Z ), Z1 Z , M ( R / Z ), sup( v) Z1 Y S sup( ) R / Z Theorem 1. If then 1. h L ( R / Z ), || h || 1, sup( h) Y 2 2 2. f ( ) h | f | ||ˆ || 2 S 3. inf | f | / || f || 0 ( S , Z1 ) not RB 2 S 4. this happens if 2 ‘contains’ a point measure New Results Definition Given a triplet topological group, and ( X , ,V ) where X is a compact : Z X is a homomorphism, V is an open neighborhood of the the identity in 1 we call Z ( X , ,V ) (V ) a Kronecker set. X, Remark Characteristic functions of Kronecker sets are uniformly recurrent points in the Bebutov system [Beb40] ( , ), {0,1} , ( s)( k ) s(k 1) Z This notion coincides with almost periodic in [GH55]. Corollary 1. If then Z1 is a Kronecker set and S ( S , Z1 ) is not a Riesz pair. is a fat Cantor set New Results Z1 Z is syndetic if there exists n N Z Z1 {0,1,2,..., n 1}, thick if n N m Z m {0,1,2,..., n 1} Z1, Definitions A subset and piecewise syndetic if it is the intersection of a syndetic and a thick set [F81]. Theorem 1.23 in [F81] page 34. If is a partition then one of the Z Z1 Z n Z i is piecewise syndetic. Observation in proof of Theorem 1.24 in [F81] page 35. If Z i is piecewise syndetic then the orbit closure of Z i contains the characteristic function of a syndetic set. Theorem 2. B satisfies Feichtinger’s conjecture iff is a Riesz pair for some syndetic (almost per.) Z1 ( , Z1 ) References J. Anderson, Extreme points in sets of positive linear maps on B(H), J. Func. Anal. 31(1979), 195-217. M. V. Bebutov, On dynamical systems in the space of continuous functions, Bull. Mos. Gos. Univ. Mat. 2 (1940). S. Bochner, Lectures on Fourier Integrals, Princeton University Press, 1959. H. Bohr, Zur Theorie der fastperiodischen Funktionen I,II,III. Acta Math. 45(1925),29-127;46(1925),101-214;47(1926),237-281 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. J. Bourgain and L. Tzafriri, On a problem of Kadison and Singer, J. Reine Angew. Math. 420(1991), 1-43 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. References 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. P. G. Casazza and R. Vershynin, Kadison-Singer meets Bourgain-Tzafriri, preprint www.math.ucdavis.edu/~vershynin/papers/kadison-singer.pdf References O. Christenson, An Introduction to Frames and Riesz Bases, Birkhauser, 2003. 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. H. Halpern, V. Kaftal, and G. Weiss,The relative Dixmier property in discrete crossed products, J. Funct. Anal. 69 (1986), 121-140. 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. N. Weaver, The Kadison-Singer problem in discrepancy theory, preprint