Hyperrigidity of C*-correspondences

TL;DR

This paper establishes a key equivalence between hyperrigidity and non-degeneracy of the Katsura ideal's action on a C*-correspondence, impacting the Hao-Ng isomorphism problem for crossed products.

Contribution

It provides a new characterization of hyperrigidity in C*-correspondences and links it to the Hao-Ng isomorphism problem under group actions.

Findings

Hyperrigidity is equivalent to non-degeneracy of the Katsura ideal's action.

Positive solutions for the Hao-Ng isomorphism problem for reduced crossed products.

Partial solutions for the Hao-Ng isomorphism problem for full crossed products.

Abstract

We show that hyperrigidity for a C*-correspondence $(A,X)$ is equivalent to non-degeneracy of the left action of the Katsura ideal $\mathcal{J}_X$ on $X$. Due to the work of Katsoulis and Ramsey, our result shows that if $G$ is a locally compact group acting on $(A,X)$ and the Katsura ideal $\mathcal{J}_X$ acts on $X$ non-degenerately then the Hao-Ng isomorphism problem for reduced crossed products has a positive solution and the Hao-Ng isomorphism problem for full crossed products has a partial solution.