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Operator algebras and subproduct systems arising from stochastic matrices.

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Title: Operator algebras and subproduct systems arising from stochastic matrices.
Authors: Dor-On, Adam1 adamd@math.bgu.ac.il, Markiewicz, Daniel1 danielm@math.bgu.ac.il
Source: Journal of Functional Analysis. Aug2014, Vol. 267 Issue 4, p1057-1120. 64p.
Subjects: Operator algebras, Stochastic matrices, Adjoint operators (Quantum mechanics), Isomorphism (Mathematics), Automorphisms, Mathematical proofs
Abstract: Abstract: We study subproduct systems in the sense of Shalit and Solel arising from stochastic matrices on countable state spaces, and their associated operator algebras. We focus on the non-self-adjoint tensor algebra, and Viselter's generalization of the Cuntz–Pimsner C*-algebra to the context of subproduct systems. Suppose that X and Y are Arveson–Stinespring subproduct systems associated to two stochastic matrices over a countable set Ω, and let and be their tensor algebras. We show that every algebraic isomorphism from onto is automatically bounded. Furthermore, and are isometrically isomorphic if and only if X and Y are unitarily isomorphic up to a *-automorphism of . When Ω is finite, we prove that and are algebraically isomorphic if and only if there exists a similarity between X and Y up to a *-automorphism of . Moreover, we provide an explicit description of the Cuntz–Pimsner algebra in the case where Ω is finite and the stochastic matrix is essential. [Copyright &y& Elsevier]
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Database: Engineering Source
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Abstract:Abstract: We study subproduct systems in the sense of Shalit and Solel arising from stochastic matrices on countable state spaces, and their associated operator algebras. We focus on the non-self-adjoint tensor algebra, and Viselter's generalization of the Cuntz–Pimsner C*-algebra to the context of subproduct systems. Suppose that X and Y are Arveson–Stinespring subproduct systems associated to two stochastic matrices over a countable set Ω, and let and be their tensor algebras. We show that every algebraic isomorphism from onto is automatically bounded. Furthermore, and are isometrically isomorphic if and only if X and Y are unitarily isomorphic up to a *-automorphism of . When Ω is finite, we prove that and are algebraically isomorphic if and only if there exists a similarity between X and Y up to a *-automorphism of . Moreover, we provide an explicit description of the Cuntz–Pimsner algebra in the case where Ω is finite and the stochastic matrix is essential. [Copyright &y& Elsevier]
ISSN:00221236
DOI:10.1016/j.jfa.2014.05.004