Tensor algebras and subproduct systems arising from

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Main Goal Paper details, We will discuss some of the results of the following paper. Dor On M 14 Adam Dor On and Daniel Markiewicz, Operator algebras and subproduct systems arising from. stochastic matrices,J Funct Anal 267 2014 no 4 pp 1057 1120. General Problem, We can encode several objects into operator algebras especially using. subproduct systems of W correspondences, How much information can we recover from the algebras.
Daniel Markiewicz Tensor algebras and stochastic matrices OAOT 2014 at ISI Bangalore 2 21. Basic framework Correspondences and subproduct systems. W modules and W correspondences,Definition, Let M be a von Neumann algebra A right M module E is called a. Hilbert W module if it is endowed with a map h i E E M such. that for all 0 E and m M,it is M linear in the second variable. h i 0 and h i 0 0, E is complete with respect to the norm k kE k h i 2 kM. it is self dual i e for every bounded M linear functional f E M. there exists f E such that f h f i, The set L E of adjointable M linear operators on E is a also W algebra. We say that E is a W correspondence when in addition E is has a left. multiplication by M given by a normal homomorphism M L E. Daniel Markiewicz Tensor algebras and stochastic matrices OAOT 2014 at ISI Bangalore 3 21. Basic framework Correspondences and subproduct systems. Hilbert spaces are W correspondences over M C, Finite graph correspondences Given a graph G G0 G1 with d.
vertices we define M Md C to be the set of diagonal matrices. and EG A Md C Aij 0 if i j 6 G1 The left right, actions are given by usual multiplication and inner product is. hA Bi Diag A B,Let M B H be a vN algebra and let M M be a unital. normal completely positive map Let M B M H be the, minimal Stinespring dilation of The Arveson Stinespring. W correspondence of is the correspondence over M 0 given by. Arv T B H M H x T T x x, with operations as follows for every T S Arv a M 0. T a T a a T I a T hT Si T S, Daniel Markiewicz Tensor algebras and stochastic matrices OAOT 2014 at ISI Bangalore 4 21.
Basic framework Correspondences and subproduct systems. Definition Shalit Solel 09 Bhat Mukherjee 10,Let M be a vN algebra let X Xn n N be a family of. W correspondences over M and let U Um n Xm Xn Xm n be. a family of bounded M linear maps We say that X is a subproduct. system over M if for all m n p N,2 Um n is co isometric. 3 The family U behaves like multiplication Um 0 and U0 n are the. right left multiplications and,Um n p Um n Ip Um n p Im Un p. When Um n is unitary for all m n we say that X is a product system. Bhat Mukherjee 10 case M C under the name inclusion systems. Product systems of Hilbert spaces were first defined by Arveson when. studying semigroups of endomorphisms of B H, Daniel Markiewicz Tensor algebras and stochastic matrices OAOT 2014 at ISI Bangalore 5 21. Basic framework Correspondences and subproduct systems. Product systems P E Given a W correspondence E over M define. P0E M PnE E and let Um m E E E be the,canonical unitary embodying associativity.
Standard Finite dimensional Hilbert space fibers Suppose that. X Xn n N is a family of fin dim Hilbert spaces such that. Xm n Xm Xn standard,Let Um n Xm Xn Xm n be the projection Then X is a. subproduct system,Theorem Muhly Solel 02 Solel Shalit 09. Let M be a vN algebra Suppose that M M is a unital normal CP. map and let Xn Arv n Then there is a canonical family of. multiplication maps U Um n for which X is a subproduct system. Daniel Markiewicz Tensor algebras and stochastic matrices OAOT 2014 at ISI Bangalore 6 21. Basic framework Tensor Toeplitz and Cuntz Pimsner algebras. Given a subproduct system X U we define the Fock W correspondence. Define for every Xm the shift operator S L FX, We shall consider several natural operator algebras associated to X U. Tensor algebra T X Alg S Xm m,not self adjoint,Toeplitz algebra T X C T X. Cuntz Pimsner algebra O X T X J X, Viselter 12 suggested the following ideal for subproduct systems let Qn.
denote the orthogonal projection onto the nth summand of Fock module. J X T T X lim kT Qn k 0, Daniel Markiewicz Tensor algebras and stochastic matrices OAOT 2014 at ISI Bangalore 7 21. Basic framework Some examples and leaving C,Example Product system P C. Let E M C and let X P C be the associated product system. We have FX n N C 2 N and T P C is closed algebra,generated by the unilateral shift Hence. T P C is the original Toeplitz algebra,Theorem Viselter 12. If E is a correspondence and its associated product system XE is faithful. then O P E O E, So the algebras for subproduct systems generalize the case of single.
correspondences via the associated product system, Daniel Markiewicz Tensor algebras and stochastic matrices OAOT 2014 at ISI Bangalore 8 21. Comparing tensor algebras Some known results, Q How much does the tensor algebra remember of the original structure. Let G and G0 be countable directed graphs, Solel 04 T P EG and T P EG0 are isometrically isomorphic if. and only if G and G0 are isomorphic as directed graphs. Kribs Katsoulis 04 T P EG and T P EG0 are boundedly. isomorphic if and only if G and G0 are isomorphic as directed graphs. Furthermore if G G0 have no sinks or sources algebraic. isomorphisms are bounded,Theorem Davidson Ramsey Shalit 11. Let X and Y be standard subproduct systems with fin dim Hilbert space. fibers Then T X is isometrically isomorphic to T Y if and only if X. and Y are unitarily isomorphic, Similar results for multivariable dyn systems Davidson Katsoulis 11.
C dynamical systems Davidson Kakariadis 12 and many more. Daniel Markiewicz Tensor algebras and stochastic matrices OAOT 2014 at ISI Bangalore 9 21. Stochastic matrices Comparing Tensor algebras,Definition. Given a countable possibly infinite set a stochastic matrix over is a. function P R such that,Pij 0 for all i j,Arv P and T P for P stochastic. There is a 1 1 correspondence between unital normal CP maps of. and stochastic matrices over given by,P f i Pij f j. Therefore a stochastic matrix P gives rise to,A subproduct system Arv P Arv P. A tensor algebra T P T Arv P, Daniel Markiewicz Tensor algebras and stochastic matrices OAOT 2014 at ISI Bangalore 10 21.
Stochastic matrices Comparing Tensor algebras,Theorem Dor On Markiewicz 14. Let P be a stochastic matrix over a state space Then up to. isomorphism of subproduct systems we have,Arv P n aij i j aij 0 if P n ij 0 aij 2. where acts as multiplication by diagonals on the left and on the. right and inner product is hA Bi Diag A B and subproduct maps are. given by s,X P m ik P n kj,Um n A B ij aik bkj, Daniel Markiewicz Tensor algebras and stochastic matrices OAOT 2014 at ISI Bangalore 11 21. Stochastic matrices Different notions of equivalence. Suppose that P Q are stochastic matrices over and T P T Q. What can we say about the associated relation between P and Q. What is the suitable version of equivalence, We have several natural isomorphism relations for tensor algebras of. stochastic matrices,Algebraic isomorphism,Bounded isomorphism.
Isometric isomorphism,Completely isometric isomorphism. Completely bounded isomorphism, However the situation turns out to be much simpler for stochastic. Daniel Markiewicz Tensor algebras and stochastic matrices OAOT 2014 at ISI Bangalore 12 21. Stochastic matrices Main results on Tensor Algebras. Theorem Dor On M 14 Automatic Continuity, Let P and Q be stochastic matrices over If T P T Q is. algebraic isomorphism then it is bounded, Remark Tensor algebras are not semi simple in general see. Davidson Katsoulis 11 so not a consequence of general machinery. The proof uses an automatic continuity lemma due to Sinclair which has. become a stepping stone for many similar results in a variety of contexts. Theorem Dor On M 14,Let P and Q be stochastic matrices over TFAE.
1 There is an isometric isomorphism of T P onto T Q. 2 there is a graded comp isometric isomorphism T P onto T Q. 3 Arv P and Arv Q are unitarily isomorphic up to change of base. Furthermore if P and Q are recurrent i e n P n ii for all i. those conditions hold if and only if P and Q are the same up to. permutation of, Daniel Markiewicz Tensor algebras and stochastic matrices OAOT 2014 at ISI Bangalore 13 21. Stochastic matrices Main results on Tensor Algebras. Recall that a stochastic matrix P is essential if for every i Pijn 0 for. some n implies that m such that Pjim 0, We also say that the support of P is the matrix supp P given by. Theorem Dor On M 14, Let P and Q be finite stochastic matrices over TFAE. 1 There is an algebraic isomorphism of T P onto T Q. 2 there is a graded comp bounded isomorphism T P onto T Q. 3 Arv P and Arv Q are similar up to change of base. Furthermore if P and Q are essential those conditions hold if and only if. P and Q have the same supports up to permutation of. Daniel Markiewicz Tensor algebras and stochastic matrices OAOT 2014 at ISI Bangalore 14 21. Stochastic matrices Summary with an example, So when P and Q are finite there are only two types of isomorphism. isometric iso classes graded completely isometric iso classes. completely isometric iso classes,algebraic iso classes bounded iso classes.
completely bounded iso classes,For every r 0 12 let. it is an essential and recurrent matrix since Pr2 Pr. Then T Pr and T Ps are,algebraically isomorphic for every r s. only isometrically isomorphic for r s, Daniel Markiewicz Tensor algebras and stochastic matrices OAOT 2014 at ISI Bangalore 15 21. Tensor algebras and subproduct systems arising from stochastic matrices Daniel Markiewicz Ben Gurion Univ of the Negev Joint Work with Adam Dor On Univ of Waterloo OAOT 2014 at ISI Bangalore Daniel Markiewicz Tensor algebras and stochastic matrices OAOT 2014 at ISI Bangalore 1 21 Main Goal Paper details We will discuss some of the results of the following paper Dor On M 14Adam

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