## 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.

Original language | English |
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Pages (from-to) | 185-227 |

Number of pages | 43 |

Journal | Integral Equations and Operator Theory |

Volume | 88 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jun 2017 |

Externally published | Yes |

### Bibliographical note

Funding Information:We would like to thank Orr Shalit for his many helpful remarks and suggestions on a draft version of this paper. This work was partially supported by the ISF within the ISF-UGC joint research program framework (Grant No. 1775/14). The work of the first author was also partially supported by an Ontario Trillium Scholarship.

Publisher Copyright:

© 2017, Springer International Publishing.

## Keywords

- Boundary representations
- C*-Envelope
- Classification
- Cuntz–Pimsner algebra
- Stochastic matrix

## ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory