Rokhlin Dimension and Inductive Limit Actions on AF-algebras

TL;DR

The paper addresses the problem of understanding crossed products arising from inductive-limit actions of finitely generated abelian groups on separable AF-algebras under finite Rokhlin dimension with commuting towers. It develops a local description framework using profinite completions and topological joins, culminating in a sequentially split embedding of $A\rtimes_{\alpha} G$ into a limit of fibered algebras, $B$, whose fibers are finite-dimensional. It then derives concrete structural consequences, showing $\dim_{nuc}(A\rtimes_{\alpha} G)\le d+r$, $\mathrm{dr}(A\rtimes_{\alpha} G)\le d+r$, and $\mathrm{sr}(A\rtimes_{\alpha} G)\le d+r+1$ (with $\mathrm{sr}\le d+1$ for $r=1$), and establishes special cases where $A\rtimes_{\alpha} \mathbb{Z}$ is an $A\mathbb{T}$-algebra and, for simple $A$, the crossed product is simple with real rank zero. The approach extends Rokhlin-dimension techniques to residually finite groups and provides a workable local-to-global framework for analyzing crossed products of AF-algebras by inductive-limit actions.

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 $α\in Aut(A)$ is approximately inner and has the Rokhlin property, we conclude that $A \rtimes_α \mathbb{Z}$ is an A$\mathbb{T}$-algebra.

Theorems & Definitions (38)

• Theorem A
• Definition 1.1
• Theorem 1.2
• Lemma 1.3
• proof
• Definition 1.4
• Definition 1.5
• Theorem 1.6
• Theorem 1.7
• proof