Constructing minimal homeomorphisms on point-like spaces and a dynamical presentation of the Jiang-Su algebra

TL;DR

This paper provides a dynamical framework to understand the Jiang-Su algebra by constructing minimal homeomorphisms on point-like spaces and representing it as a groupoid C*-algebra, linking dynamical systems and operator algebras.

Contribution

It introduces a novel dynamical presentation of the Jiang-Su algebra via an étale equivalence relation and minimal homeomorphisms on spaces with trivial cohomology.

Findings

Constructed minimal homeomorphisms on infinite, compact metric spaces with trivial cohomology.

Established an étale equivalence relation whose groupoid C*-algebra is isomorphic to the Jiang-Su algebra.

Connected dynamical systems properties with the structure of the Jiang-Su algebra.

Abstract

The principal aim of this paper is to give a dynamical presentation of the Jiang-Su algebra. Originally constructed as an inductive limit of prime dimension drop algebras, the Jiang-Su algebra has gone from being a poorly understood oddity to having a prominent positive role in George Elliott's classification programme for separable, nuclear C*-algebras. Here, we exhibit an etale equivalence relation whose groupoid C*-algebra is isomorphic to the Jiang-Su algebra. The main ingredient is the construction of minimal homeomorphisms on infinite, compact metric spaces, each having the same cohomology as a point. This construction is also of interest in dynamical systems. Any self-map of an infinite, compact space with the same cohomology as a point has Lefschetz number one. Thus, if such a space were also to satisfy some regularity hypothesis (which our examples do not), then the Lefschetz-Hopf Theorem would imply that it does not admit a minimal homeomorphism.