Diagonal-preserving graded isomorphisms of Steinberg algebras

TL;DR

This paper characterizes when diagonal-preserving graded isomorphisms exist between Steinberg algebras, using groupoid reconstruction, and applies these results to Leavitt path and graph $C^*$-algebras.

Contribution

It provides a new characterization of diagonal-preserving graded isomorphisms for Steinberg algebras via groupoid reconstruction, extending to graph-related algebras.

Findings

Reconstruction of groupoids from Steinberg algebras.

Characterization of diagonal-preserving graded isomorphisms.

Applications to Leavitt path and graph $C^*$-algebras.

Abstract

We study Steinberg algebras constructed from ample Hausdorff groupoids over commutative integral domains with identity. We reconstruct (graded) groupoids from (graded) Steinberg algebras and use this to characterise when there is a diagonal-preserving (graded) isomorphism between two (graded) Steinberg algebras. We apply this characterisation to groupoids of directed graphs in order to study diagonal-preserving (graded) isomorphisms of Leavitt path algebras and graph $C^*$-algebras.