Operator algebras and subproduct systems arising from stochastic matrices

Adam Dor-On, Daniel Markiewicz

Research output: Contribution to journalArticlepeer-review


We study subproduct systems in the sense of Shalit and Solel arising from stochastic matrices on countable state spaces, and their associated operator algebras. We focus on the non-self-adjoint tensor algebra, and Viselter's generalization of the Cuntz-Pimsner C*-algebra to the context of subproduct systems. Suppose that X and Y are Arveson-Stinespring subproduct systems associated to two stochastic matrices over a countable set Ω, and let T+(X) and T+(Y) be their tensor algebras. We show that every algebraic isomorphism from T+(X) onto T+(Y) is automatically bounded. Furthermore, T+(X) and T+(Y) are isometrically isomorphic if and only if X and Y are unitarily isomorphic up to a *-automorphism of ℓ(Ω). When Ω is finite, we prove that T+(X) and T+(Y) are algebraically isomorphic if and only if there exists a similarity between X and Y up to a *-automorphism of ℓ(Ω). Moreover, we provide an explicit description of the Cuntz-Pimsner algebra O(X) in the case where Ω is finite and the stochastic matrix is essential.

Original languageEnglish
Pages (from-to)1057-1120
Number of pages64
JournalJournal of Functional Analysis
Issue number4
StatePublished - 15 Aug 2014
Externally publishedYes

Bibliographical note

Funding Information:
The first author was partially supported by GIF (German–Israeli Foundation) research grant No. 2297-2282.6/201 , and the second author was partially supported by grant 2008295 from the U.S.–Israel Binational Science Foundation .


  • Cuntz-Pimsner algebra
  • Stochastic matrix
  • Subproduct system
  • Tensor algebra

ASJC Scopus subject areas

  • Analysis


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