Ideal structure of C*-algebras of commuting local homeomorphisms

TL;DR

This paper characterizes the primitive ideal space of a broad class of groupoid C*-algebras, including 2-graph and topological graph C*-algebras, unifying various known results on their ideal structures.

Contribution

It introduces the notion of harmonious families of bisections to analyze the ideal structure of C*-algebras from commuting local homeomorphisms, extending existing theories.

Findings

Determines the primitive ideal space for a large class of groupoid C*-algebras.

Unifies ideal structure results for various classes of C*-algebras.

Provides a framework applicable to 2-graph and topological graph C*-algebras.

Abstract

We determine the primitive ideal space and hence the ideal lattice of a large class of separable groupoid C*-algebras that includes all 2-graph C*-algebras. A key ingredient is the notion of harmonious families of bisections in etale groupoids associated to finite families of commuting local homeomorphisms. Our results unify and recover all known results on ideal structure for crossed products of commutative C*-algebras by free abelian groups, for graph C*-algebras, and for Katsura's topological graph C*-algebras.