Rokhlin Dimension and Inductive Limit Actions on AF-algebras

TL;DR

This paper studies actions with finite Rokhlin dimension on AF-algebras, providing a local description of the crossed product and showing that certain automorphisms lead to A$ ext{T}$-algebras.

Contribution

It introduces a local description of crossed products for actions with finite Rokhlin dimension and characterizes when these crossed products are A$ ext{T}$-algebras.

Findings

Crossed products have a local structure described by the paper.

Automorphisms with the Rokhlin property yield A$ ext{T}$-algebras as crossed products.

Finite Rokhlin dimension with commuting towers influences the structure of the crossed product.

Abstract

Given a separable, AF-algebra A and an inductive limit action on A of a finitely generated abelian group with finite Rokhlin dimension with commuting towers, we give a local description of the associated crossed product C*-algebra. In particular, when A is unital and $\alpha \in Aut(A)$ is approximately inner and has the Rokhlin property, we conclude that $A \rtimes_{\alpha} \mathbb{Z}$ is an A$\mathbb{T}$-algebra.