Rokhlin dimension for actions of residually compact groups

TL;DR

This paper introduces Rokhlin dimension for residually compact group actions on C*-algebras, extending previous notions and showing how it preserves key properties like nuclear dimension and absorption under crossed products.

Contribution

It defines Rokhlin dimension for residually compact groups and demonstrates its implications for nuclear dimension, absorption, and stability of crossed products.

Findings

Finite nuclear dimension is preserved under crossed products with finite Rokhlin dimension.

Absorption of strongly self-absorbing C*-algebras is preserved under such crossed products.

Crossed products are stable if the group contains a non-open cocompact closed subgroup.

Abstract

We introduce the concept of Rokhlin dimension for actions of residually compact groups on C*-algebras, which extends and unifies previous notions for actions of compact groups, residually finite groups and the reals. We then demonstrate that finite nuclear dimension (respectively, absorption of a strongly self-absorbing C*-algebra) is preserved under the formation of crossed products by residually compact group actions with finite Rokhlin dimension (respectively, finite Rokhlin dimension with commuting towers). Furthermore, if second countable residually compact group contains a non-open cocompact closed subgroup, then crossed products arising from actions with finite Rokhlin dimension are stable. Finally, we study the relationship between the tube dimension of a topological dynamical system and the Rokhlin dimension of the induced C*-dynamical system.