Rokhlin Property for Group Actions on Hilbert $C^*$-modules

TL;DR

This paper develops Rokhlin properties for discrete group actions on Hilbert $C^*$-modules and bimodules, showing how these properties induce similar actions on related algebras and studying their permanence under crossed products.

Contribution

It introduces Rokhlin properties for group actions on $C^*$-correspondences and Hilbert bimodules, and analyzes their effects on associated algebras and permanence properties.

Findings

Group actions with Rokhlin property induce similar actions on associated $C^*$-algebras.

Permanence of nuclear dimension and $ ext{D}$-absorption under crossed products.

Investigation of outerness for Hilbert bimodules.

Abstract

We introduce Rokhlin properties for certain discrete group actions on $C^*$-correspondences as well as on Hilbert bimodules and analyze them. It turns out that the group actions on any $C^*$-correspondence $E$ with Rokhlin property induces group actions on the associated $C^*$-algebra $\mathcal O_E$ with Rokhlin property and the group actions on any Hilbert bimodule with Rokhlin property induces group actions on the linking algebra with Rokhlin property. Permanence properties of several notions such as nuclear dimension and $\mathcal D$-absorbing property with respect to crossed product of Hilbert $C^*$-modules with groups, where group actions have Rokhlin property, are studied. We also investigate a notion of outerness for Hilbert bimodules.