On von Neumann regularity of ample groupoid algebras

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.