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) and T+(Y) be their tensor algebras. We show that every algebraic isomorphism from T+(X) onto T+(Y) is automatically bounded. Furthermore, T+(X) and T+(Y) are isometrically isomorphic if and only if X and Y are unitarily isomorphic up to a *-automorphism of ℓ∞(Ω). When Ω is finite, we prove that T+(X) and T+(Y) are algebraically isomorphic if and only if there exists a similarity between X and Y up to a *-automorphism of ℓ∞(Ω). Moreover, we provide an explicit description of the Cuntz-Pimsner algebra O(X) in the case where Ω is finite and the stochastic matrix is essential.
Bibliographical noteFunding Information:
The first author was partially supported by GIF (German–Israeli Foundation) research grant No. 2297-2282.6/201 , and the second author was partially supported by grant 2008295 from the U.S.–Israel Binational Science Foundation .
- Cuntz-Pimsner algebra
- Stochastic matrix
- Subproduct system
- Tensor algebra
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