Tensor algebras of C∗-correspondences and their C∗-envelopes

By Elias G. Katsoulis, David W. Kribs. 2005

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Abstract We show that the C ∗ -envelope of the tensor algebra of an arbitrary C ∗ -correspondence X coincides with the Cuntz–Pimsner algebra O X , as defined by Katsura [T. Katsura, On C ∗ -algebras associated with C ∗ -correspondences, J. Funct. Anal. 217 (2004) 366–401]. This improves earlier results of Muhly and Solel [P.S. Muhly, B. Solel, Tensor algebras over C ∗ -correspondences: Representations, dilations and C ∗ -envelopes, J. Funct. Anal. 158 (1998) 389–457] and Fowler, Muhly and Raeburn [N. Fowler, P. Muhly, I. Raeburn, Representations of Cuntz–Pimsner algebras, Indiana Univ. Math. J. 52 (2003) 569–605], who came to the same conclusion under the additional hypothesis that X is strict and faithful.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/de431e40c699fc7383a26370e4f6fc23660e501c

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Citations in the corpus

Operator Algebras for Multivariable Dynamics (2007, methodology): “By a result of Katsoulis and Kribs [23], O(E) is the C*-envelope of T+(E).”
Semicrossed Products of Operator Algebras by Semigroups (2014, unknown): “This remark has multiple interpretations: firstly it suggests a generalization of the work of Katsoulis and Kribs [62] on C*-correspondences, secondly it fits exactly into the framework of nonselfadjoint algebras, and thirdly it fits in Arveson’s program on the C*-envelope [7, 61].”
Dilation theory, commutant lifting and semicrossed products (2011, background): “The C*-envelope is the Cuntz-Pimsner algebra O(E) [42, 29, 34].”