Equivalent groupoids have Morita equivalent Steinberg algebras

TL;DR

This paper proves that equivalent Hausdorff ample groupoids have Morita equivalent Steinberg algebras, and applies this to show certain graph modifications do not change the Morita equivalence class of associated Leavitt path algebras.

Contribution

It establishes Morita equivalence of Steinberg algebras for equivalent groupoids and links graphical graph operations to Morita equivalence of Leavitt path algebras.

Findings

Equivalent groupoids have Morita equivalent Steinberg algebras.

Graph collapsing does not change Morita equivalence class of Leavitt path algebras.

Graphical constructions for Morita equivalent $C^*$-algebras also apply to Leavitt path algebras.

Abstract

Let $G$ and $H$ be Hausdorff ample groupoids and let $R$ be a commutative unital ring. We show that if $G$ and $H$ are equivalent in the sense of Muhly-Renault-Williams, then the associated Steinberg algebras of locally constant $R$-valued functions with compact support are Morita equivalent. We deduce that collapsing a ``collapsible subgraph" of a directed graph in the sense of Crisp and Gow does not change the Morita-equivalence class of the associated Leavitt path $R$-algebra, and therefore a number of graphical constructions which yield Morita equivalent $C^*$-algebras also yield Morita equivalent Leavitt path algebras.