Rokhlin dimension: duality, tracial properties, and crossed products

TL;DR

This paper investigates compact group actions with finite Rokhlin dimension, exploring their duals, ideal structures, and how such actions preserve various properties of C*-algebras, including nuclearity and the UCT, with new insights even for the Rokhlin property.

Contribution

It generalizes the understanding of Rokhlin dimension, characterizes dual actions, and shows preservation of key C*-algebra classes under these actions, including new results for the Rokhlin property.

Findings

Duals of actions with finite Rokhlin dimension are characterized.

Crossed products preserve several classes of C*-algebras under these actions.

Finite Rokhlin dimension with commuting towers implies the weak tracial Rokhlin property.

Abstract

We study compact group actions with finite Rokhlin dimension, particularly in relation to crossed products. For example, we characterize the duals of such actions, generalizing previous partial results for the Rokhlin property. As an application, we determine the ideal structure of their crossed products. Under the assumption of so-called commuting towers, we show that taking crossed products by such actions preserves a number of relevant classes of C*-algebras, including: $D$-absorbing C*-algebras, where $D$ is a strongly self-absorbing C*-algebra, stable C*-algebras, C*-algebras with finite nuclear dimension (or decomposition rank), C*-algebras with finite stable rank (or real rank), and C*-algebras whose K-theory is either trivial, rational, or $n$-divisible for $n\in\mathbb{N}$. The combination of nuclearity and the UCT is also shown to be preserved by these actions. Some of these results are new even in the well-studied case of the Rokhlin property. Additionally, and under some technical assumptions, we show that finite Rokhlin dimension with commuting towers implies the (weak) tracial Rokhlin property. At the core of our arguments is a certain local approximation of the crossed product by a continuous $C(X)$-algebra with fibers that are stably isomorphic to the underlying algebra. The space $X$ is computed in some cases of interest, and we use its description to construct a $\mathbb{Z}_2$-action on a unital AF-algebra and on a unital Kirchberg algebra satisfying the UCT, whose Rokhlin dimensions with and without commuting towers are finite but do not agree.