Advertisement

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
Article

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 \(C({\mathbb {T}}, M_d({\mathbb {C}}))\), 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.

Keywords

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

Mathematics Subject Classification

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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

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

References

  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. https://math.berkeley.edu/~arveson/Dvi/unExt.pdf (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. http://arxiv.org/abs/1505.06773 [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

Personalised recommendations