The Ideal Structure of Steinberg Algebras

TL;DR

This paper investigates the ideal structure of Steinberg algebras linked to ample groupoids, demonstrating that all ideals can be derived from isotropy group algebras, thus providing an algebraic perspective on the Effros-Hahn conjecture.

Contribution

It develops a theory of induced ideals in crossed product algebras and proves that all ideals are intersections of ideals from isotropy group algebras, extending the understanding of Steinberg algebra structures.

Findings

Every ideal in the crossed product algebra is an intersection of induced ideals from isotropy groups.

The work provides an algebraic analogue of the Effros-Hahn conjecture.

Application to Steinberg algebras associated with ample groupoids.

Abstract

Given an ample action of an inverse semigroup on a locally compact and Hausdorff topological space, we study the ideal structure of the crossed product algebra associated with it. By developing a theory of induced ideals, we manage to prove that every ideal in the crossed product algebra may be obtained as the intersection of ideals induced from isotropy group algebras. This can be interpreted as an algebraic version of the Effros-Hahn conjecture. Finally, as an application of our result, we study the ideal structure of a Steinberg algebra associated with an ample groupoid by interpreting it as an inverse semigroup crossed product algebra.