Representations of $(A,B)$ C*-correspondences

Alonso Delfín

Published 2022-08-31Version 1

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 and prove the existence of representations of C*-correspondences on a pair of Hilbert spaces. 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.