In this talk we will discuss C*-envelopes of tensor algebras associated to finite irreducible stochastic matrices. We show that there are examples of such C*-envelopes that are non-*-isomorphic to either the Toeplitz or the Cuntz-Pimsner algebras, which was somewhat unexpected.

We provide a detailed characterization of the boundary representations of the tensor algebra inside the Toeplitz algebra, and apply K-theory and extension theory to classify all such C*-envelopes in terms of the underlying stochastic matrices, up to stable isomorphism and *-isomorphism.

This is joint work with Daniel Markiewicz.