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Rokhlin dimension for actions of residually compact groups

Xin Cao, Xiaochun Fang, Jianchao Wu

TL;DR

This work defines Rokhlin dimension for actions of residually compact groups on C*-algebras, unifying prior notions for compact, residually finite, and real-group actions. It formulates the dimension via equivariant order-zero maps from $C(G/H)$ into central-sequence algebras, parameterized by residually compact approximations, and establishes core transfer principles. The authors prove that finite Rokhlin dimension (even without commuting towers) preserves finite nuclear dimension and $D$-absorption under crossed products, and they prove stability when a non-open cocompact subgroup exists; they also connect Rokhlin dimension to tube dimension in topological dynamics. These results broaden the class of group actions for which regularity properties transfer to crossed products, with implications for the classification program and structural analysis of non-discrete dynamical systems.

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.

Rokhlin dimension for actions of residually compact groups

TL;DR

This work defines Rokhlin dimension for actions of residually compact groups on C*-algebras, unifying prior notions for compact, residually finite, and real-group actions. It formulates the dimension via equivariant order-zero maps from into central-sequence algebras, parameterized by residually compact approximations, and establishes core transfer principles. The authors prove that finite Rokhlin dimension (even without commuting towers) preserves finite nuclear dimension and -absorption under crossed products, and they prove stability when a non-open cocompact subgroup exists; they also connect Rokhlin dimension to tube dimension in topological dynamics. These results broaden the class of group actions for which regularity properties transfer to crossed products, with implications for the classification program and structural analysis of non-discrete dynamical systems.

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.
Paper Structure (13 sections, 44 theorems, 146 equations)

This paper contains 13 sections, 44 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some basic notations
  4. Central sequence algebras
  5. Order zero maps
  6. Asymptotic dimension
  7. Residually compact groups
  8. Box spaces and asymptotic dimension
  9. Rokhlin dimension for actions of residually compact groups
  10. Nuclear dimension of crossed products
  11. $D$-absorption of crossed products
  12. Stability of crossed products
  13. Tube dimension vs. Rokhlin dimension

Key Result

Theorem B

Let $A$ be a separable C*-algebra, let $G$ be a second countable residually compact group, and let $\alpha\colon G\to {\rm Aut}(A)$ be a continuous action. Let $\sigma=(G_n)_{n\in \mathbb{N}}$ be a regular residually compact approximation of $G$. Then the following inequality holds:

Theorems & Definitions (93)

  • Definition A: see Definition \ref{['def main']} for the general case
  • Theorem B: see Theorem \ref{['thm:dimnuc']}
  • Theorem C: see Theorem \ref{['thm: D-ab']}
  • Theorem D: see Theorem \ref{['thm: stable']}
  • Theorem E: see Theorem \ref{['thm: tubedim']}
  • Remark 2.1
  • Remark 2.2: Kir06
  • Lemma 2.3: HSWW17
  • Definition 2.4
  • Proposition 2.5
  • ...and 83 more