Representations of C*-correspondences on pairs of Hilbert spaces

Abstract

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.

AMSLaTeX; 19 pages. V4 fixes an issue with Remark 4.7 and updates the statement in Theorem 4.8 accordingly. To appear in Journal of Operator Theory

Cited by in corpus (2)

• C*-like modules and matrix $p$-operator norms
• $L^p$-modules and $L^p$-correspondences