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Contributions to the theory of $ \mathrm{C}^*$-correspondences with applications to multivariable dynamics


Authors: Evgenios T. A. Kakariadis and Elias G. Katsoulis
Journal: Trans. Amer. Math. Soc. 364 (2012), 6605-6630
MSC (2010): Primary 47L55, 47L40, 46L05, 37B20
DOI: https://doi.org/10.1090/S0002-9947-2012-05627-3
Published electronically: July 17, 2012
MathSciNet review: 2958949
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Abstract | References | Similar Articles | Additional Information

Abstract: Motivated by the theory of tensor algebras and multivariable $ \mathrm {C}^*$-dynamics, we revisit two fundamental techniques in the theory of $ \mathrm {C}^*$-corres-
pondences, the ``addition of a tail'' to a non-injective $ \mathrm {C}^*$-correspondence and the dilation of an injective $ \mathrm {C}^*$-correspondence to an essential Hilbert bimodule. We provide a very broad scheme for ``adding a tail'' to a non-injective $ \mathrm {C}^*$-correspondence; our scheme includes the ``tail'' of Muhly and Tomforde as a special case. We illustrate the diversity and necessity of our tails with several examples from the theory of multivariable $ \mathrm {C}^*$-dynamics. We also exhibit a transparent picture for the dilation of an injective $ \mathrm {C}^*$-correspondence to an essential Hilbert bimodule. As an application of our constructs, we prove two results in the theory of multivariable dynamics that extend earlier results. We also discuss the impact of our results on the description of the $ \mathrm {C}^*$-envelope of a tensor algebra as the Cuntz-Pimsner algebra of the associated $ \mathrm {C}^*$-correspondence.


References [Enhancements On Off] (What's this?)

  • 1. W. Arveson, Notes on the unique extension property, 2006, http://math.berkeley.edu/ arveson/Dvi/unExt.pdf.
  • 2. T. Bates, J. Hong, I. Raeburn and W. Szymanski, The ideal structure of the $ \mathrm {C}^*$-algebras of infinite graphs, Illinois J. Math. 46 (2002), 1159-1176. MR 1988256 (2004i:46105)
  • 3. D. P. Blecher and C. Le Merdy, Operator algebras and their modules--an operator space approach, volume 30 of London Mathematical Society Monographs, New Series, The Clarendon Press, Oxford University Press, Oxford, 2004. MR 2111973 (2006a:46070)
  • 4. J. Cuntz, K-theory for certain $ \mathrm {C}^*$-algebras. II, J. Operator Theory 5 (1981), 101-108. MR 613050 (84k:46053)
  • 5. K. Davidson and E. Katsoulis, Operator algebras for multivariable dynamics, Mem. Amer. Math. Soc. 209 (2011), no. 983. MR 2752983
  • 6. K. Davidson and E. Katsoulis, Semicrossed products of the disc algebra, Proc. Amer. Math. Soc. 140 (2012), 3479-3484.
  • 7. K. Davidson and J. Roydor, $ \mathrm {C}^*$-envelopes of tensor algebras for multivariable dynamics, Proc. Edinb. Math. Soc. (2) 53 (2010), 333-351. MR 2653236
  • 8. V. Deaconu, A. Kumjian, D. Pask and A. Sims, Graphs of $ C^*$-correspondences and Fell bundles, Indiana Univ. Math. J. 59 (2010).
  • 9. M.A. Dritschel and S.A. McCullough, Boundary representations for families of representations of operator algebras and spaces, J. Operator Theory, 53(1) (2005), 159-167. MR 2132691 (2006a:47095)
  • 10. N. Fowler, P. Muhly and I. Raeburn, Representations of Cuntz-Pimsner algebras, Indiana Univ. Math. J. 52 (2003), 569-605. MR 1986889 (2005d:46114)
  • 11. N. Fowler and I. Raeburn, The Toeplitz algebra of a Hilbert bimodule, Indiana Univ. Math. J. 48 (1999), 155-181. MR 1722197 (2001b:46093)
  • 12. M. Hamana, Injective envelopes of operator systems, Publ. RIMS Kyoto Univ. 15(1979), 773-785. MR 566081 (81h:46071)
  • 13. E. T.A. Kakariadis, Semicrossed products and reflexivity, J. Operator Theory 67 (2012), 379-395.
  • 14. E. T.A. Kakariadis and E.G. Katsoulis, Semicrossed products of operator algebras and their $ C^*$-envelopes, J. Funct. Anal. 262 (2012), 3108-3124. MR 2885949
  • 15. E. T.A. Kakariadis and E.G. Katsoulis, manuscript.
  • 16. E. Katsoulis and D. Kribs, Isomorphisms of algebras associated with directed graphs, Math. Ann. 330, (2004), 709-728. MR 2102309 (2005i:47114)
  • 17. E. G. Katsoulis and D. Kribs, Tensor algebras of $ \mathrm {C}^*$-correspondences and their $ \mathrm {C}^*$-envelopes, J. Funct. Anal. 234(1) (2006), 226-233. MR 2214146 (2007a:46061)
  • 18. T. Katsura, A construction of $ C\sp *$-algebras from $ \mathrm {C}^*$-correspondences, Contemp. Math. 335 (2003), 173-182. MR 2029622 (2005k:46131)
  • 19. T. Katsura, On $ C\sp *$-algebras associated with $ \mathrm {C}^*$-correspondences, J. Funct. Anal. 217(2) (2004), 366-401. MR 2102572 (2005e:46099)
  • 20. C. Lance, Hilbert $ \mathrm {C}^*$-modules. A toolkit for operator algebraists, London Mathematical Society Lecture Note Series, 210 Cambridge University Press, Cambridge, 1995. x+130 pp. ISBN: 0-521-47910-X. MR 1325694 (96k:46100)
  • 21. P.S. Muhly and B. Solel, Tensor algebras over $ \mathrm {C}^*$-correspondences: representations, dilations and $ \mathrm {C}^*$-envelopes J. Funct. Anal. 158 (1998), 389-457. MR 1648483 (99j:46066)
  • 22. P.S. Muhly and B. Solel, On the simplicity of some Cuntz-Pimsner algebras, Math. Scand. 83 (1998), 53-73. MR 1662076 (99m:46140)
  • 23. P. S. Muhly and M. Tomforde, Adding tails to $ \mathrm {C}^*$-correspondences, Doc. Math. 9 (2004), 79-106. MR 2054981 (2005a:46117)
  • 24. W. Paschke, The crossed product of a $ \mathrm {C}^*$-algebra by an endomorphism Proc. Amer. Math. Soc. 80 (1980), 113-118. MR 574518 (81m:46081)
  • 25. G. K. Pedersen, $ C^{\ast } $-algebras and their automorphism groups, volume 14 of London Mathematical Society Monographs, Academic Press Inc.,1979. MR 548006 (81e:46037)
  • 26. J. Peters, Semicrossed products of C*-algebras, J. Funct. Anal. 59 (1984), 498-534. MR 769379 (86e:46063)
  • 27. J. Peters, The $ \mathrm {C}^*$-envelope of a semicrossed product and nest representations, Contemp. Math. 503 (2009), 197-215. MR 2590624 (2011c:47168)
  • 28. M. Pimsner, A class of $ C\sp *$-algebras generalizing both Cuntz-Krieger algebras and crossed products by $ \mathbb{Z}$, Free probability theory (Waterloo, ON, 1995), 189-212, Fields Inst. Commun., 12, Amer. Math. Soc., Providence, RI, 1997. MR 1426840 (97k:46069)
  • 29. G. Popescu, Non-commutative disc algebras and their representations, Proc. Amer. Math. Soc. 124, (1996), 2137-2148. MR 1343719 (96k:47077)
  • 30. I. Raeburn, Graph algebras, CBMS Regional Conference Series in Mathematics, 103, 2005. MR 2135030 (2005k:46141)
  • 31. I. Raeburn and W. Szymanski, Cuntz-Krieger algebras of infinite graphs and matrices, Trans. Amer. Math. Soc. 356 (2004), 39-59. MR 2020023 (2004i:46087)
  • 32. J. Schweizer, Dilations of $ C^*$-correspondences and the simplicity of Cuntz-Pimsner algebras, J. Funct. Anal. 180(2) (2000), 404-425. MR 1814994 (2002f:46113)
  • 33. P. Stacey, Crossed products of $ \mathrm {C}^*$-algebras by $ \ast $-endomorphisms, J. Austr. Math. Soc., Ser. A 56 (1993), 204-212. MR 1200792 (94a:46077)

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Additional Information

Evgenios T. A. Kakariadis
Affiliation: Department of Mathematics, University of Athens, 15784 Athens, Greece
Email: mavro@math.uoa.gr

Elias G. Katsoulis
Affiliation: Department of Mathematics, University of Athens, 15784 Athens, Greece
Address at time of publication: Department of Mathematics, East Carolina University, Greenville, North Carolina 27858
Email: katsoulise@ecu.edu

DOI: https://doi.org/10.1090/S0002-9947-2012-05627-3
Keywords: $\mathrm{C}^{*}$-correspondences, $\mathrm{C}^{*}$-envelope, adding a tail, Hilbert bimodule, crossed product by endomorphism
Received by editor(s): April 19, 2011
Published electronically: July 17, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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