Integral Equations and Operator Theory

, Volume 88, Issue 2, pp 185–227 | Cite as

C*-Envelopes of Tensor Algebras Arising from Stochastic Matrices

  • Adam Dor-OnEmail author
  • Daniel Markiewicz


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.


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

Mathematics Subject Classification

Primary 47L30 46L55 46L35 Secondary 46L80 60J10 


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We would like to thank Orr Shalit for his many helpful remarks and suggestions on a draft version of this paper.


  1. 1.
    Arveson, W.B.: Subalgebras of \(C^{\ast } \)-algebras. Acta Math. 123, 141–224 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arveson, W.B.: Subalgebras of \(C^{\ast } \)-algebras. Acta Math. II 128(3–4), 271–308 (1972)CrossRefzbMATHGoogle Scholar
  3. 3.
    Arveson, W.B.: An Invitation to \(C^*\)-Algebras. Graduate Texts in Mathematics, vol. 39. Springer, New York (1976)CrossRefzbMATHGoogle Scholar
  4. 4.
    Arveson, W.B.: Notes on extensions of \(C^{^*}\)-algebras. Duke Math. J. 44(2), 329–355 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Arveson, W.B.: Notes on the Unique Extension Property. (2006)
  6. 6.
    Arveson, W.B.: The noncommutative Choquet boundary. J. Am. Math. Soc. 21(4), 1065–1084 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Arveson, W.B.: The noncommutative Choquet boundary II: hyperrigidity. Isr. J. Math. 184, 349–385 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Blackadar, B.: \(K\)-Theory for Operator Algebras, vol. 5, 2nd edn. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  9. 9.
    Blecher, D.P., Ruan, Z.J., Sinclair, A.M.: A characterization of operator algebras. J. Funct. Anal. 89(1), 188–201 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Brown, L.G., Dadarlat, M.: Extensions of \(C^\ast \)-algebras and quasidiagonality. J. Lond. Math. Soc. (2) 533, 582–600 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cuntz, J., Krieger, W.: A class of \(C^{\ast } \)-algebras and topological Markov chains. Invent. Math. 56, 251 (1980). doi: 10.1007/BF01390048 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cuntz, J.: A class of \(C^{\ast } \)-algebras and topological Markov chains. II. Reducible chains and the ext-functor for \(C^{\ast }\)-algebras. Invent. Math. 63, 25 (1981). doi: 10.1007/BF01389192 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Davidson, K.R., Kennedy, M.: The Choquet boundary of an operator system. Duke Math. J. 164(15), 2989–3004 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Davidson, K.R., Ramsey, C., Shalit, O.M.: The isomorphism problem for some universal operator algebras. Adv. Math. 228(1), 167–218 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dixmier, J.: \(C^*\)-Algebras. North-Holland Publishing Co., Amsterdam (1977). (Translated from the French by Francis Jellett, North-Holland Mathematical Library, vol. 15)zbMATHGoogle Scholar
  16. 16.
    Dor-On, A., Markiewicz, D.: Operator algebras and subproduct systems arising from stochastic matrices. J. Funct. Anal. 267(4), 1057–1120 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Durrett, R.: Probability: Theory and Examples. Cambridge Series in Statistical and Probabilistic Mathematics, 4th edn. Cambridge University Press, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  18. 18.
    Dritschel, M.A., McCullough, S.A.: Boundary representations for families of representations of operator algebras and spaces. J. Oper. Theory 53(1), 159–167 (2005)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Eilers, S., Loring, T.A., Pedersen, G.K.: Morphisms of extensions of \(C^*\)-algebras: pushing forward the Busby invariant. Adv. Math. 147(1), 74–109 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Eilers, S., Restorff, G., Ruiz, E.: Classification of extensions of classifiable \(C^\ast \)-algebras. Adv. Math. 222(6), 2153–2172 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Eilers, S., Restorff, G., Ruiz, E., Sørensen, A.P.W.: Geometric Classification of Unital Graph C*-Algebras of Real Rank Zero. [math.OA] (2015)
  22. 22.
    Hamana, M.: Injective envelopes of operator systems. Publ. Res. Inst. Math. Sci. 15(3), 773–785 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kakariadis, E.T.A.: The Šilov boundary for operator spaces. Integral Equ. Oper. Theory 76(1), 25–38 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kakariadis, E.T.A., Shalit, O.: On Operator Algebras Associated with Monomial Ideals in Noncommuting Variables. arXiv:1501.06495 [math.OA] (2015)
  25. 25.
    Katsoulis, E.G., Kribs, D.W.: Tensor algebras of \(C^*\)-correspondences and their \(C^*\)-envelopes. J. Funct. Anal. 234(1), 226–233 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Katsura, T.: On \(C^*\)-algebras associated with \(C^*\)-correspondences. J. Funct. Anal. 217(2), 366–401 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kumjian, A., Pask, D., Raeburn, I.: Cuntz–Krieger algebras of directed graphs. Pac. J. Math. 184(1), 161–174 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lance, E.C.: Hilbert \(C^*\)-Modules. London Mathematical Society Lecture Note Series, vol. 210. Cambridge University Press, Cambridge (1995). (A toolkit for operator algebraists)CrossRefGoogle Scholar
  29. 29.
    Manuilov, V.M., Troitsky, E.V.: Hilbert \(C^*\)-Modules, Translations of Mathematical Monographs, vol. 226, American Mathematical Society, Providence, RI, 2005, Translated from the 2001 Russian original by the authorsGoogle Scholar
  30. 30.
    Muhly, P.S., Solel, B.: Tensor algebras over \(C^*\)-correspondences: representations, dilations, and \(C^*\)-envelopes. J. Funct. Anal. 158(2), 389–457 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Muhly, P.S., Solel, B.: On the Morita equivalence of tensor algebras. Proc. Lond. Math. Soc. (3) 81(1), 113–168 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Paschke, W.L.: Inner product modules over \(B^{\ast } \)-algebras. Trans. Am. Math. Soc. 182, 443–468 (1973)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Paschke, W.L., Salinas, N.: Matrix algebras over \(O_{n}\). Michigan Math. J. 26(1), 3–12 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Pimsner, M.V.: A class of \(C^*\)-algebras generalizing both Cuntz–Krieger algebras and crossed products by \({ Z}\), Free probability theory. Waterloo (1995) (Fields Inst. Commun., vol. 12, Am. Math. Soc. Providence, RI 1997, pp. 189–212)Google Scholar
  35. 35.
    Raeburn, I.: Graph algebras, CBMS Regional Conference Series in Mathematics, vol. 103, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, (2005)Google Scholar
  36. 36.
    Raeburn, I., Szymański, W.: Cuntz–Krieger algebras of infinite graphs and matrices. Trans. Am. Math. Soc 356(1), 39–59 (2004). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Raeburn, I., Williams, D.P.: Morita Equivalence and Continuous-Trace \(C^*\)-Algebras. Mathematical Surveys and Monographs, vol. 60. American Mathematical Society, Providence (1998)CrossRefzbMATHGoogle Scholar
  38. 38.
    Rørdam, M., Larsen, F., Laustsen, N.: An Introduction to \(K\)-Theory for \(C^*\)-Algebras. London Mathematical Society Student Texts, vol. 49. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  39. 39.
    Seneta, E.: Non-negative Matrices and Markov Chains. Springer Series in Statistics. Springer, New York (2006). (revised reprint of the second (1981) edition. Springer, New York)zbMATHGoogle Scholar
  40. 40.
    Shalit, O.M., Solel, B.: Subproduct systems. Doc. Math. 14, 801–868 (2009)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Viselter, A.: Covariant representations of subproduct systems. Proc. Lond. Math. Soc. (3) 102(4), 767–800 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Viselter, A.: Cuntz–Pimsner algebras for subproduct systems. Int. J. Math. 23(8), 1250081 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Voiculescu, D.: A non-commutative Weyl-von Neumann theorem. Rev. Roumaine Math. Pures Appl. 21(1), 97–113 (1976)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Pure Mathematics DepartmentUniversity of WaterlooWaterlooCanada
  2. 2.Department of MathematicsBen-Gurion University of the NegevBe’er ShevaIsrael

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