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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
Published electronically: July 17, 2012
MathSciNet review: 2958949
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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.


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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: http://dx.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.