C*-Envelopes of Tensor Algebras Arising from Stochastic Matrices

Abstract

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.

C*-envelopes for operator algebras with a coaction and co-universal C*-algebras for product systems

“…One specific noncommutative boundary is the noncommutative analogue of the Shilov boundary, called the C*-envelope, which can be thought of as the smallest C*-algebra containing the given operator algebra in a reasonable sense. Computing the C*-envelope in various cases has been of interest and use to many authors over the years [17,18,21,22,32,39,49]. C*-envelopes have also had recent applications in classification of nonselfadjoint operator algebras [19], finite dimensional approximation [11], crossed products [39], group theory [6,34], noncommutative geometry [12], and noncommutative convexity [24].…”

Doob equivalence and non-commutative peaking for Markov chains

“…For the basic theory of operator on Hilbert space and their algebras we refer the reader to [3,11,13]. In this paper we resolve problems related to Markov chains motivated from studying operator algebras associated to stochastic matrices as in [14,15]. Definition 1.1.…”

“…In [15], completely peaking states of P were computed under the assumption of multiple arrival (see [15,Corollary 3.14]). Based on this, the C * -envelope of T + (P ) is computed and classified (see [15,Section 3] and [15, Theorem 5.5 & Theorem 5.6]).…”

Gysin sequences and SU(2)‐symmetries of C∗‐algebras

“…In particular, we provide, for our class of algebras, a partial answer to Open Question 3 in [40, Section 6] concerning the computation of the ‐theory groups of the Cuntz–Pimsner and Toeplitz–Pimsner algebras of a subproduct system. Note in this respect that the paper [16] also contains valuable computations of ‐theory groups relating to Viselter's question. The present text offers a completely new approach, which exploits topological features like the existence of higher dimensional Gysin sequences.…”

“…The theory of subproduct systems was further developed by Viselter, who extended the notions of covariant representation and of Cuntz–Pimsner algebras of a ‐correspondence to this more general framework [39, 40]. More recently, Dor‐On and Markiewicz [15, 16] applied the theory of subproduct systems to the study of stochastic matrices.…”