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 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...

Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball

“…In fact, one also looks at subproduct systems over more general semigroups, and it is useful to allow fibers that are Hilbert W*-correspondences, and not just Hilbert spaces, but such generality is beyond the scope of the present work. Subproduct systems give rise to a class of natural operator algebras, and in recent years these algebras have been investigated by several researchers [9,23,25,26,35,43,84,85]. We will now explain how algebras of bounded analytic functions on homogeneous varieties are operator algebras associated with subproduct systems, and indicate points of intersection with previous works.…”

“…In [23,25] the algebras A X and L X were classified in terms of the structure of the subproduct systems. In [78,Proposition 7.4] (see also [23,Proposition 3.1]) it was shown that if X and Y are subproduct subsystems of (C d ) ⊗n n∈N , then X and Y are isomorphic, if and only if I X is obtained from I Y by unitary change of variables.…”

“…Finally, we mention that in [43, Theorem 3.4] (following work done in [25]) it was shown that operator algebras A X and A Y (arising from subproduct systems X and Y ) are boundedly isomorphic if and only if X and Y are similar. We leave it for future work to parse what this means in terms of bounded isomorphisms between algebras of the form A(V), where V ⊂ B d is a homogeneous variety.…”

C*-Envelopes of Tensor Algebras Arising from Stochastic Matrices

“…There has been important work on the operator algebras arising from subproduct systems over C, or equivalently, the special case of subproduct systems whose C*-correspondence fibers are actually Hilbert spaces, see for example [SS09,DRS11,KS15]. In our previous paper [DOM14], we turned to the simplest case for which the fibers of the subproduct system are not Hilbert spaces. Namely, we considered the case of subproduct systems of C*-correspondences over ℓ ∞ (Ω) when Ω is countable with more than one point.…”

“…Such a subproduct system and its associated operator algebras are conveniently parametrized by a stochastic matrix P over the state space Ω. In [DOM14], we considered isomorphism problems of the tensor algebras associated to stochastic matrices, via these subproduct systems. stochastic matrix P over Ω P , the ideal Ker(π P ) in the sequence ( * ) is *-isomorphic to a direct sum of n P ≤ |Ω P | copies of the algebra of compact operators.…”