Contributions to the theory of 𝐶*-correspondences with applications to multivariable dynamics

Kakariadis, Evgenios; Katsoulis, Elias

doi:10.1090/S0002-9947-2012-05627-3

Contributions to the theory of $\mathrm {C}^*$-correspondences with applications to multivariable dynamics
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by Evgenios T. A. Kakariadis and Elias G. Katsoulis PDF

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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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