Green's theorem for crossed products by Hilbert $C^*$-bimodules

TL;DR

This paper extends Green's theorem to the setting where a group acts on a $C^*$-algebra via a Hilbert bimodule, establishing a Morita equivalence in this new context.

Contribution

It introduces a Green's theorem analogue for actions generated by Hilbert $C^*$-bimodules instead of automorphisms, broadening the theorem's applicability.

Findings

Established Morita equivalence for crossed products by Hilbert bimodules

Generalized Green's theorem to the case G=Z with bimodule actions

Provided a new framework for analyzing $C^*$-algebra actions

Abstract

Green's theorem gives a Morita equivalence $C_0(G/H,A)\rtimes G\sim A\rtimes H$ for a closed subgroup $H$ of a locally compact group $G$ acting on a $C^*$-algebra $A$. We prove an analogue of Green's theorem in the case $G=\mathbb{Z}$, where the automorphism generating the action is replaced by a Hilbert $C^*$-bimodule.