Ideals of \'etale groupoid algebras with coefficients in a sheaf with applications to topological dynamics

TL;DR

This paper proves the Effros-Hahn conjecture for groupoid algebras with sheaf coefficients, characterizes their ideal structure, and applies these results to topological dynamics and C*-algebra simplicity.

Contribution

It extends the Effros-Hahn conjecture to sheaf-coefficient groupoid algebras and links algebraic properties to dynamical features of inverse semigroup actions.

Findings

Characterization of ideals in skew inverse semigroup rings.

Criteria for von Neumann regularity, primitivity, and simplicity of groupoid algebras.

Conditions for simplicity of the associated C*-algebra.

Abstract

We prove the Effros-Hahn conjecture for groupoid algebras with coefficients in a sheaf, obtaining as a consequence a description of the ideals in skew inverse semigroup rings. We also use the description of the ideals to characterize when the groupoid algebras with coefficients in a sheaf are von Neumann regular, primitive, semiprimitive, or simple. We apply our results to the topological dynamics of actions of inverse semigroups, describing the existence of dense orbits and minimality in terms of primitivity and simplicity, respectively, of the associated algebra. Moreover, we apply our results to the usual complex groupoid algebra of continuous functions with compact support, used to build the C*-algebra associated with a groupoid, and describe criteria for its simplicity.