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.1 2 ADAM DOR-ON AND DANIEL MARKIEWICZ is easy to find two graphs with d vertices and irreducible adjacency matrix whose C*-algebras are not isomorphic (see [Rae05]). In contrast, in section 5 we show that if X is the Arveson-Stinespring subproduct system of a d × d irreducible stochastic matrix, then O(X) ∼ = C(T) ⊗ M d (C). More generally, we also provide an explicit description for the Cuntz-Pimsner algebra of a subproduct system associated to essential finite stochastic matrices.On the other hand, the non-self-adjoint tensor algebra T + (E) of a C*-correspondence E over A has often proven to be a strong invariant of the correspondence. Muhly and Solel [MS00] proved that if E and F are aperiodic C*-correspondences, then T + (E) is isometrically isomorphic to T + (F ) if and only if E and F are isometrically isomorphic as A-bimodules. Similarly, Katsoulis and Kribs [KK04] and Solel [Sol04] proved that if G and G ′ are countable directed graphs, then the tensor algebras T + (G) and T + (G ′ ) are isomorphic as algebras if and only if G and G ′ are isomorphic as directed graphs. See also Davidson and Katsoulis [DK11] for another important example of this phenomenon of increased acuity of the normed (non-self-adjoint) algebras as opposed to C*algebras, perhaps first recognized in Arveson [Arv67] and Arveson and Josephson [AJ69]. The tensor algebras of subproduct systems were first considered by Solel and Shalit [SS09] in the special case of M = C, and they analyzed the problem of graded isomorphism of their tensor algebras. The general isomorphism problem for such subproduct systems was resolved by Davidson, Ramsey and Shalit [DRS11]. They proved that if X, Y are subproduct systems of finitedimensional Hilbert space fibers, then T + (X) and T + (Y ) are isometrically isomorphic if and only if X and Y are (unitarily) isomorphic.On the other hand, the recent work of Gurevich [Gur12] provides a useful contrast. Although in this paper we focus on subproduct systems over N, it is possible to consider mo...

In this paper we study the C*-envelope of the (non-self-adjoint) tensor algebra associated via subproduct systems to a finite irreducible stochastic matrix P .Firstly, we identify the boundary representations of the tensor algebra inside the Toeplitz algebra, also known as its non-commutative Choquet boundary. As an application, we provide examples of C*-envelopes that are not *-isomorphic to either the Toeplitz algebra or the Cuntz-Pimsner algebra. This characterization required a new proof for the fact that the Cuntz-Pimsner algebra associated to P is isomorphic to C(T, M d (C)), filling a gap in a previous paper.We then proceed to classify the C*-envelopes of tensor algebras of stochastic matrices up to *isomorphism and stable isomorphism, in terms of the underlying matrices. This is accomplished by determining the K-theory of these C*-algebras and by combining this information with results due to Paschke and Salinas in extension theory. This classification is applied to provide a clearer picture of the various C*-envelopes that can land between the Toeplitz and the Cuntz-Pimsner algebras.

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