C*-Envelopes of Tensor Algebras Arising from Stochastic Matrices

Adam Dor-On, Daniel Markiewicz

Research output: Contribution to journalArticlepeer-review

10 Scopus citations


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, Md(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 languageEnglish
Pages (from-to)185-227
Number of pages43
JournalIntegral Equations and Operator Theory
Issue number2
StatePublished - 1 Jun 2017


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

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory


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