Journal of Operator Theory

Volume 92, Issue 1, Summer 2024 pp. 167-188.

Representations of $C^*$-correspondences on pairs of Hilbert spaces

Authors: Alonso Delfin
Author institution: Department of Mathematics, University of Oregon, Eugene OR 97403-1222, U.S.A. \textit{and} Department of Mathematics, University of Colorado, Boulder CO 80309-0395, U.S.A.

Summary: We study representations of Hilbert bimodules on pairs of Hilbert spaces. If $A$ is a $C^*$-algebra and $\mathsf{X}$ is a right Hilbert $A$-module, we use such representations to faithfully represent the $C^*$-algebras $\mathcal{K}_A(\mathsf{X})$ and $\mathcal{L}_A(\mathsf{X})$. We then extend this theory to define representations of $(A,B)$ $C^*$-correspondences on a pair of Hilbert spaces and show how these can be obtained from any nondegenerate representation of $B$. As an application of such representations, we give necessary and sufficient conditions on an $(A,B)$ $C^*$-correspondences to admit a Hilbert $A$-$B$-bimodule structure. Finally, we show how to represent the interior tensor product of two $C^*$-correspondences

DOI: http://dx.doi.org/10.7900/jot.2022sep02.2431
Keywords: $C^*$-correspondences, Hilbert bimodules, representations, adjointable maps, interior tensor product

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