Rokhlin dimension and C*-dynamics

TL;DR

This paper introduces the concept of Rokhlin dimension for group actions on C*-algebras, generalizing the Rokhlin property, and demonstrates its prevalence and implications for structural properties like nuclear dimension and Z-stability.

Contribution

It develops the notion of Rokhlin dimension for finite group actions on C*-algebras, extending the classical Rokhlin property and exploring its genericity and preservation of structural properties.

Findings

Finite Rokhlin dimension is generic for automorphisms of Z-stable C*-algebras.

Crossed products by automorphisms with finite Rokhlin dimension preserve finite nuclear dimension.

Automorphisms from minimal homeomorphisms have finite Rokhlin dimension.

Abstract

We develop the concept of Rokhlin dimension for integer and for finite group actions on C*-algebras. Our notion generalizes the so-called Rokhlin property, which can be thought of as Rokhlin dimension 0. We show that finite Rokhlin dimension is prevalent and appears in cases in which the Rokhlin property cannot be expected: the property of having finite Rokhlin dimension is generic for automorphisms of Z-stable C*-algebras, where Z denotes the Jiang-Su algebra. Moreover, crossed products by automorphisms with finite Rokhlin dimension preserve the property of having finite nuclear dimension, and under a mild additional hypothesis also preserve Z-stability. In topological dynamics our notion may be interpreted as a topological version of the classical Rokhlin lemma: automorphisms arising from minimal homeomorphisms of finite dimensional compact metrizable spaces always have finite Rokhlin dimension. The latter result has by now been generalized by Szabo to the case of free and aperiodic integer (in fact, Z^d) actions on compact metrizable and finite dimensional spaces.