Extending Pure States on C*-Algebras and Feichtinger’s Conjecture

1. Extending Pure States on C*-Algebras and Feichtinger’s Conjecture Wayne Lawton Department of Mathematics National University of Singapore matwml@nus.edu.sg http://www.math.nus.edu.sg/~matwml Special Program on Operator Algebras5th Asian Mathematical Conference Putra World Trade Centre, Kuala Lumpur MALAYSIA 22-26 June 2009

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

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

4. Synthesis Operator denotes a complex Hilbert space and the Hermitian form is linear in and congugate-linear in A subset defines a Synthesis Operator

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

6. is a Frame for if it is a Bessel set Frames that satisfies any of the following equivalent conditions: 1. is surjective, 2. is injective, 3. Proof of Equivalence: [Chr03], pages 102-103. Example. is a Bessel set, but not a frame for Proof: [Chr03], 98-99.

7. is a Riesz Set if it is a Bessel set that satisfies Riesz Sets any of the following equivalent conditions: 1. is bijective, is bijective, 2. 3. Proof of Equivalence: [Chr03], 66-68, 123-125. Example: Union of n > 1 Riesz bases for is always a frame for but never a Riesz set. Remark: is the grammian, and is the dual- grammian used by Amos Ron and Zuowei Shen http://www.math.nus.edu.sg/~matzuows/publist.html

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

9. If is stationary set then is a Stationary Sets 1. Bessel set iff there exists a symbol function and then is a 2. Frame iff 3. Tight Frame iff is constant on its support 4. Riesz set iff Proof. [Chr], 143-145.

10. with symbol Representation as Exponentials Stationary Bessel Sets Representation as Translates

11. Two Conjectures Definition A Fechtinger set is a finite union of Riesz sets - Riesz set is one satisfying Feichtinger Conjecture: Every Bessel set is Feichtinger set Definition Let An - Conjecture: For every every Riesz set is a finite union of - Riesz sets

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

13. A State on a unital - algebra is a linear functional States on C*-Algebras that satisfies any of the following equiv. cond. 1. 2. is convex and weakly compact A Pure State is an extremal state Krein-Milman  Examples

14. Does every pure state on The Kadison-Singer Problem have a unique extension to a state on Remarks Problem arose from Dirac quantization Hahn-Banach extensions always exist YES answer to KS is equivalent to: - combination of the Feichtinger and conjectures - Paving Conjecture: every is pave-able - other conjectures in mathematics and engineering

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

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

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

18. New Results Definition Given a triplet where is a compact topological group, is a homomorphism, and is an open neighborhood of the the identity in a Kronecker set. we call Remark Characteristic functions of Kronecker sets are uniformly recurrent points in the Bebutov system [Beb40] This notion coincides with almost periodic in [GH55]. Corollary 1. If is a Kronecker set and is a fat Cantor set then is not a Riesz pair.

19. New Results Definitions A subset is syndetic if there exists thick if 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 is 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.)

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