Minimal dynamical systems on connected odd dimensional spaces

TL;DR

This paper proves that certain crossed product C*-algebras arising from minimal homeomorphisms on odd-dimensional spheres and related spaces have rational tracial rank at most one, making them classifiable.

Contribution

It establishes the classification of crossed product C*-algebras from minimal dynamical systems on odd-dimensional spaces, including real projective spaces.

Findings

Crossed products from minimal homeomorphisms on spheres have rational tracial rank ≤ 1.

Classifiability of these C*-algebras is achieved.

Results extend to minimal systems on spaces with specific cohomological properties.

Abstract

Let $\beta: S^{2n+1}\to S^{2n+1}$ be a minimal homeomorphism ($n\ge 1$). We show that the crossed product $C(S^{2n+1})\rtimes_{\beta} \Z$ has rational tracial rank at most one. More generally, let $\Omega$ be a connected compact metric space with finite covering dimension and with $H^1(\Omega, \Z)=\{0\}.$ Suppose that $K_i(C(\Omega))=\Z\oplus G_i$ for some finite abelian group $G_i,$ $i=0,1.$ Let $\beta: \Omega\to\Omega$ be a minimal homeomorphism. We also show that $A=C(\Omega)\rtimes_{\beta}\Z$ has rational tracial rank at most one and is classifiable. In particular, this applies to the minimal dynamical systems on odd dimensional real projective spaces. This was done by studying the minimal homeomorphisms on $X\times \Omega,$ where $X$ is the Cantor set.