Crossed products and minimal dynamical systems

TL;DR

This paper proves that minimal dynamical systems on infinite compact metric spaces with finite covering dimension have crossed product $C^*$-algebras classified by their Elliott invariants, and shows these algebras have generalized tracial rank at most one under certain conditions.

Contribution

It establishes classification results for crossed product $C^*$-algebras arising from minimal homeomorphisms on infinite compact metric spaces, extending to mean dimension zero systems.

Findings

Crossed products are isomorphic iff their Elliott invariants are isomorphic.

Crossed products with mean dimension zero have generalized tracial rank at most one.

Such crossed products belong to a classifiable class of amenable simple $C^*$-algebras.

Abstract

Let $X$ be an infinite compact metric space with finite covering dimension and let $\alpha, \beta : X\to X$ be two minimal homeomorphisms. We prove that the crossed product $C^*$-algebras $C(X)\rtimes_\alpha\Z$ and $C(X)\rtimes_\belta\Z$ are isomorphic if and only if they have isomorphic Elliott invariant. In a more general setting, we show that if $X$ is an infinite compact metric space and if $\alpha: X\to X$ is a minimal homeomorphism such that $(X, \alpha)$ has mean dimension zero, then the tensor product of the crossed product with a UHF-algebra of infinite type has generalized tracial rank at most one. This implies that the crossed product is in a classifiable class of amenable simple $C^*$-algebras.