The Hao-Ng isomorphism theorem for reduced crossed products

TL;DR

This paper establishes an isomorphism between reduced crossed products and Cuntz-Pimsner algebras for group actions on C*-correspondences, extending the Hao-Ng theorem to a broader setting.

Contribution

It proves the Hao-Ng isomorphism for reduced crossed products in the context of locally compact Hausdorff groups acting on C*-correspondences.

Findings

The isomorphism ${ m O}_{X times_r G} o { m O}_X times_r G$ is established.

The result applies to non-degenerate C*-correspondences with generalized gauge actions.

The theorem extends previous results to locally compact Hausdorff groups.

Abstract

We prove the Hao-Ng isomorphism for reduced crossed products by locally compact Hausdorff groups. More precisely, for a non-degenerate $\mathrm{C}^*$-correspondence $X$ and a generalized gauge action $G \curvearrowright X$ by a locally compact Hausdorff group $G$, we prove the commutation ${\mathcal{O}}_{X\rtimes_rG}\cong {\mathcal{O}}_X\rtimes_rG$ of the reduced crossed product with the Cuntz-Pimsner C*-algebra construction.