Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4590309 | Journal of Functional Analysis | 2014 | 64 Pages |
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 T+(X)T+(X) and T+(Y)T+(Y) be their tensor algebras. We show that every algebraic isomorphism from T+(X)T+(X) onto T+(Y)T+(Y) is automatically bounded. Furthermore, T+(X)T+(X) and T+(Y)T+(Y) are isometrically isomorphic if and only if X and Y are unitarily isomorphic up to a *-automorphism of ℓ∞(Ω)ℓ∞(Ω). When Ω is finite, we prove that T+(X)T+(X) and T+(Y)T+(Y) are algebraically isomorphic if and only if there exists a similarity between X and Y up to a *-automorphism of ℓ∞(Ω)ℓ∞(Ω). Moreover, we provide an explicit description of the Cuntz–Pimsner algebra O(X)O(X) in the case where Ω is finite and the stochastic matrix is essential.