# 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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