On von Neumann regularity of ample groupoid algebras

TL;DR

This work provides a comprehensive characterization of von Neumann regularity for Steinberg algebras $R\mathscr{G}$ of ample groupoids and extends the analysis to graded regularity via cocycles to a group $H$. The main theorem states that $R\mathscr{G}$ is von Neumann regular if and only if $R$ is regular, $\mathscr{G}$ is a directed union of quasi-compact open subgroupoids, and the order of every finite isotropy subgroup is invertible in $R$, with a parallel reduction to the identity component for graded regularity. These results yield a complete resolution of when inverse semigroup algebras $RS$ are regular, and they recover Hazrat’s graded regularity results for Leavitt path algebras while proving new graded regularity statements for Deaconu-Renault groupoids and Nekrashevych-Exel-Pardo algebras. The paper also includes a groupoid-free appendix for the inverse semigroup case and demonstrates the broad applicability of the regularity criteria to several important classes of groupoid algebras, highlighting implications for graded $K$-theory and structural analysis.

Abstract

We completely characterize when the algebra of an ample groupoid with coefficients in an arbitrary unital ring is von Neumann regular and, more generally, when the algebra of a graded ample groupoid is graded von Neumann regular. Our main application is to resolve the question, open since 1970, of when the algebra of an inverse semigroup is von Neumann regular. As applications, we recover known results on regularity and graded regularity of Leavitt path algebras, and prove a number of new results, in particular concerning graded regularity of algebras of Deaconu-Renault groupoids and Nekrashevych-Exel-Pardo algebras of self-similar groups.

Theorems & Definitions (62)

• Proposition 2.1
• Theorem 2.2: Birkhoff
• proof
• Theorem 2.3: Connell
• Proposition 2.4: Weissglass
• proof
• Proposition 3.1
• proof
• Proposition 3.2
• proof