Group actions on Smale space C*-algebras

TL;DR

This paper investigates how group actions on Smale spaces influence the structure and properties of associated C*-algebras, demonstrating preservation of key properties under certain conditions.

Contribution

It establishes that effective actions induce strongly outer actions on homoclinic algebras and that finite Rokhlin dimension properties transfer to stable and unstable C*-algebras in irreducible Smale spaces.

Findings

Effective group actions produce strongly outer actions on homoclinic algebras.

Finite Rokhlin dimension passes from homoclinic to stable and unstable algebras.

Crossed products preserve properties like nuclear dimension and Z-stability.

Abstract

Group actions on a Smale space and the actions induced on the C*-algebras associated to such a dynamical system are studied. We show that an effective action of a discrete group on a mixing Smale space produces a strongly outer action on the homoclinic algebra. We then show that for irreducible Smale spaces, the property of finite Rokhlin dimension passes from the induced action on the homoclinic algbera to the induced actions on the stable and unstable C*-algebras. In each of these cases, we discuss the preservation of properties---such as finite nuclear dimension, Z-stability, and classification by Elliott invariants---in the resulting crossed products.