Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph C*-algebras and Leavitt path algebras

TL;DR

This paper establishes a deep connection between groupoid equivalence and stable isomorphism, and applies this to classify diagonal-preserving stable isomorphisms of graph C*-algebras and Leavitt path algebras.

Contribution

It proves that ample groupoids with sigma-compact units are equivalent iff they are stably isomorphic, linking this to graph algebra isomorphisms.

Findings

Groupoid equivalence iff stable isomorphism for ample groupoids

Diagonal-preserving stable isomorphisms induce groupoid isomorphisms

Certain Leavitt path algebras are not stably *-isomorphic

Abstract

We prove that ample groupoids with sigma-compact unit spaces are equivalent if and only if they are stably isomorphic in an appropriate sense, and relate this to Matui's notion of Kakutani equivalence. We use this result to show that diagonal-preserving stable isomorphisms of graph C*-algebras or Leavitt path algebras give rise to isomorphisms of the groupoids of the associated stabilised graphs. We deduce that the Leavitt path algebras $L_Z(E_2)$ and $L_Z(E_{2-})$ are not stably *-isomorphic.