Representations of C*-correspondences on pairs of Hilbert spaces

TL;DR

This paper develops a framework for representing C*-correspondences on pairs of Hilbert spaces, enabling faithful representations of associated C*-algebras and conditions for bimodule structures.

Contribution

It introduces a method to represent C*-correspondences on pairs of Hilbert spaces and characterizes when such correspondences admit a Hilbert bimodule structure.

Findings

Faithful representations of $ ext{K}_A( extsf{X})$ and $ ext{L}_A( extsf{X})$ are constructed.

Provides conditions for $(A,B)$ C*-correspondences to have a Hilbert bimodule structure.

Shows how to represent the interior tensor product of C*-correspondences.

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.