Multiply generated dynamical systems and the duality of higher rank graph algebras

TL;DR

This paper introduces a semidirect product groupoid for systems of local homeomorphisms, establishing its amenability and linking it to higher rank graph algebras to interpret duality in E-theory.

Contribution

It constructs a new groupoid framework for higher rank graph algebras, providing a topological perspective on duality in E-theory between associated C*-algebras.

Findings

The groupoid constructed is amenable.

The associated algebra is Cuntz-like.

Provides a topological interpretation of E-theory duality.

Abstract

We define a semidirect product groupoid of a system of partially defined local homeomorphisms $T=(T_{1},..., T_{r})$. We prove that this construction gives rise to amenable groupoids. The associated algebra is a Cuntz-like algebra. We use this construction for higher rank graph algebras in order to give a topological interpretation for the duality in $E$-theory between $C^{*}(\Lambda)$ and $C^{*}(\Lambda^{op})$.