Graded Steinberg algebras and partial actions

TL;DR

This paper establishes a connection between graded Steinberg algebras of groupoids and partial skew inverse semigroup rings, providing new insights into their structure and ideal correspondence, with applications to Leavitt path algebras.

Contribution

It introduces a novel realization of graded Steinberg algebras as partial skew inverse semigroup rings and explores their ideal structure and applications.

Findings

Steinberg algebra is graded isomorphic to partial skew group ring for partial group actions.

One-to-one correspondence between open invariant subsets and graded ideals.

Realization of partial $C^{*}$-algebras as partial skew group rings.

Abstract

Given a graded ample Hausdorff groupoid, we realise its graded Steinberg algebra as a partial skew inverse semigroup ring. We use this to show that for a partial action of a discrete group on a locally compact Hausdorff topological space, the Steinberg algebra of the associated groupoid is graded isomorphic to the corresponding partial skew group ring. We show that there is a one-to-one correspondence between the open invariant subsets of the topological space and the graded ideals of the partial skew group ring. We also consider the algebraic version of the partial $C^{*}$-algebra of an abelian group and realise it as a partial skew group ring via a partial action of the group on a topological space. Applications to the theory of Leavitt path algebras are given.