Reconstruction of graded groupoids from graded Steinberg algebras

TL;DR

This paper demonstrates how to reconstruct a graded ample Hausdorff groupoid from its graded Steinberg algebra, establishing a link between algebraic structures and topological groupoids, with implications for graph $C^*$-algebras.

Contribution

It provides a method to reconstruct graded groupoids from their Steinberg algebras and shows that diagonal-preserving isomorphisms imply $C^*$-isomorphisms for certain graph algebras.

Findings

Reconstruction of graded groupoids from Steinberg algebras.

Diagonal-preserving isomorphisms imply $C^*$-isomorphisms for graphs with exits.

Applicable over any commutative integral domain with 1.

Abstract

We show how to reconstruct a graded ample Hausdorff groupoid with topologically principal neutrally graded component from the ring structure of its graded Steinberg algebra over any commutative integral domain with 1, together with the embedding of the canonical abelian subring of functions supported on the unit space. We deduce that diagonal-preserving ring isomorphism of Leavitt path algebras implies $C^*$-isomorphism of $C^*$-algebras for graphs $E$ and $F$ in which every cycle has an exit.