Rokhlin dimension: duality, tracial properties, and crossed products

• Published in: Ergodic theory and dynamical systems Vol. 41; no. 2; pp. 408 - 460
• Main Authors: GARDELLA, EUSEBIO, HIRSHBERG, ILAN, SANTIAGO, LUIS
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.02.2021
• Subjects: Algebra; Commuting; Mathematical analysis; Original Article; Towers; C ∗ -algebra; 46L35; crossed products; 46L80; Rokhlin dimension; 46L55
• ISSN: 0143-3857; 1469-4417
• DOI: 10.1017/etds.2019.68

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 $C^{\ast }$ -algebras, including: $D$ -absorbing $C^{\ast }$ -algebras, where $D$ is a strongly self-absorbing $C^{\ast }$ -algebra; stable $C^{\ast }$ -algebras; $C^{\ast }$ -algebras with finite nuclear dimension (or decomposition rank); $C^{\ast }$ -algebras with finite stable rank (or real rank); and $C^{\ast }$ -algebras whose $K$ -theory is either trivial, rational, or $n$ -divisible for $n\in \mathbb{N}$ . The combination of nuclearity and the universal coefficient theorem (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.