C*-envelopes of tensor algebras arising from stochastic matrices

TL;DR

This paper investigates the C*-envelopes of tensor algebras linked to stochastic matrices, providing a classification based on K-theory and extension theory, and identifying boundary representations within the Toeplitz algebra.

Contribution

It offers a new classification of C*-envelopes for tensor algebras from stochastic matrices, including examples not isomorphic to known algebras, and clarifies the structure of these envelopes.

Findings

Identified boundary representations inside the Toeplitz algebra.

Provided examples of C*-envelopes not isomorphic to Toeplitz or Cuntz-Pimsner algebras.

Classified C*-envelopes up to *-isomorphism and stable isomorphism using K-theory.

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 $C(\mathbb{T}, M_d(\mathbb{C}))$, 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.