Operator algebras and subproduct systems arising from stochastic matrices

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 $\Omega$, and let $\mathcal{T}_+(X)$ and $\mathcal{T}_+(Y)$ be their tensor algebras. We show that every algebraic isomorphism from $\mathcal{T}_+(X)$ onto $\mathcal{T}_+(Y)$ is automatically bounded. Furthermore, $\mathcal{T}_+(X)$ and $\mathcal{T}_+(Y)$ are isometrically isomorphic if and only if $X$ and $Y$ are unitarily isomorphic up to a *-automorphism of $\ell^\infty(\Omega)$. When $\Omega$ is finite, we prove that $\mathcal{T}_+(X)$ and $\mathcal{T}_+(Y)$ are algebraically isomorphic if and only if there exists a similarity between $X$ and $Y$ up to a *-automorphism of $\ell^\infty(\Omega)$. Moreover, we provide an explicit description of the Cuntz-Pimsner algebra $\mathcal{O}(X)$ in the case where $\Omega$ is finite and the stochastic matrix is essential.