AT-algebras from fiberwise essentially minimal zero-dimensional dynamical systems

TL;DR

This paper introduces fiberwise essentially minimal zero-dimensional dynamical systems and proves that their associated crossed product C*-algebras are AT-algebras with classifiable K-theory, including many nontrivial non-minimal examples.

Contribution

It defines fiberwise essentially minimal systems and shows their crossed products are AT-algebras with real rank zero under certain conditions, expanding the class of known examples.

Findings

Crossed products are AT-algebras.

Systems without periodic points have real rank zero.

Many nontrivial, non-minimal examples are constructed.

Abstract

We introduce a type of zero-dimensional dynamical system (a pair consisting of a totally disconnected compact metrizable space along with a homeomorphism of that space), which we call "fiberwise essentially minimal", and we prove that the associated crossed product $ C^* $-algebra of such a system is an AT-algebra. Under the additional assumption that the system has no periodic points, we prove that the associated crossed product $ C^* $-algebra has real rank zero, which tells us that such $ C^* $-algebras are classifiable by $ K $-theory. We show that the definition of fiberwise essentially minimality allows one to produce many nontrivial examples of such systems (ones that are neither minimal nor essentially minimal). The associated crossed product $ C^* $-algebras to these nontrivial examples are of particular interest because they are non-simple (unlike in the minimal case).