The structure of crossed products by automorphisms of $C (X, D)$

TL;DR

This paper constructs large subalgebras within crossed products of $C(X, D)$ by automorphisms, establishing structural properties like $Z$-stability and real rank zero in complex examples not approachable by existing methods.

Contribution

It introduces a new construction of centrally large subalgebras in crossed products of $C(X, D)$, enabling analysis of their structural properties in challenging cases.

Findings

Proves $Z$-stability and stable rank one in specific crossed products.

Demonstrates pure infiniteness in certain examples.

Provides methods applicable when $D$ is not $Z$-stable or $X$ is infinite dimensional.

Abstract

We construct centrally large subalgebras in crossed products of $C (X, D)$ by automorphisms in which $D$ is simple, $X$ is compact metrizable, the automorphism induces a minimal homeomorphism of $X$, and a mild technical assumption holds. We use this construction to prove structural properties of the crossed product, such as (tracial) $Z$-stability, stable rank one, real rank zero, and pure infiniteness, in a number of examples. Our examples are not accessible via methods based on finite Rokhlin dimension, either because $D$ is not $Z$-stable or because $X$ is infinite dimensional.