Classification of the crossed product $C(M)\times_\theta\Z_p$ for certain pairs $(M,\theta)$

TL;DR

This paper classifies certain crossed product C*-algebras arising from homeomorphisms of low-dimensional compact spaces, showing they are isomorphic iff their quotient spaces are homeomorphic with matching fixed point sets.

Contribution

It provides a complete classification of crossed product C*-algebras for prime-period homeomorphisms on 2-dimensional compact spaces, linking algebraic isomorphism to topological orbit equivalence.

Findings

Isomorphism of crossed products characterized by quotient space homeomorphisms.

Orbit equivalence implies isomorphism of the associated crossed product C*-algebras.

Conditions on fixed point sets and cohomology determine algebraic classification.

Abstract

Let $M$ be a separable compact Hausdorff space with $\dim M\le 2$ and $\theta\colon M\to M$ be a homeomorphism with prime period $p$ ($p\ge 2$). Set $M_\theta=\{x\in M| \theta(x)=x\}\not=\varnothing$ and $M_0=M\backslash M_\theta$. Suppose that $M_0$ is dense in $M$ and $\mathrm H^2(M_0/\theta,\Z)\cong 0$, $\mathrm H^2(\chi(M_0/\theta),\Z)\cong 0$. Let $M'$ be another separable compact Hausdorff space with $\dim M'\le 2$ and $\theta'$ be the self--homeomorphism of $M'$ with prime period $p$. Suppose that $M_0'=M'\backslash M_{\theta'}'$ is dense in $M'$. Then $C(M)\times_\theta\Z_p\cong C(M')\times_{\theta'}\Z_p$ iff there is a homeomorphism $F$ from $M/\theta$ onto $M'/\theta'$ such that $F(M_\theta)=M'_{\theta'}$. Thus, if $(M,\theta)$ and $(M',\theta')$ are orbit equivalent, then $C(M)\times_\theta\Z_p\cong C(M')\times_{\theta'}\Z_p$.