Strongly graded groupoid and directed graph algebras

TL;DR

This paper establishes a connection between the strong grading properties of reduced $C^*$-algebras of graded ample groupoids and Steinberg algebras, with applications to graph algebras.

Contribution

It proves an equivalence between strong grading in $C^*$-algebras and Steinberg algebras, extending to Leavitt path and graph algebras.

Findings

Reduced $C^*$-algebra is strongly graded iff Steinberg algebra is strongly graded

Results apply to Leavitt path algebras and graph $C^*$-algebras

Provides new insights into the structure of graph-related algebras

Abstract

We show the reduced $C^*$-algebra of a graded ample groupoid is a strongly graded $C^*$-algebra if and only if the corresponding Steinberg algebra is a strongly graded ring. We apply this result to get a theorem about the Leavitt path algebra and $C^*$-algebra of an arbitrary graph.