Rokhlin dimension for flows

TL;DR

This paper introduces a new concept called Rokhlin dimension for flows on C*-algebras, extending previous notions and showing it preserves important properties like nuclear dimension and absorption under crossed products, with applications to topological flows.

Contribution

It generalizes Rokhlin property to Rokhlin dimension for flows, demonstrating preservation of structural properties and providing classification results for certain crossed product C*-algebras.

Findings

Finite Rokhlin dimension preserves nuclear dimension and absorption.

Flows from free topological flows have finite Rokhlin dimension.

Crossed products of free and minimal flows are classifiable by the Elliott invariant.

Abstract

We introduce a notion of Rokhlin dimension for one parameter automorphism groups of C*-algebras. This generalizes Kishimoto's Rokhlin property for flows, and is analogous to the notion of Rokhlin dimension for actions of the integers and other discrete groups introduced by the authors and Zacharias in previous papers. We show that finite nuclear dimension and absorption of a strongly self-absorbing C*-algebra are preserved under forming crossed products by flows with finite Rokhlin dimension, and that these crossed products are stable. Furthermore, we show that a flow on a commutative C*-algebra arising from a free topological flow has finite Rokhlin dimension, whenever the spectrum is a locally compact metrizable space with finite covering dimension. For flows that are both free and minimal, this has strong consequences for the associated crossed product C*-algebras: Those containing a non-zero projection are classified by the Elliott invariant (for compact manifolds this consists of topological K-theory together with the space of invariant probability measures and a natural pairing given by the Ruelle-Sullivan map).