Invariant means on Boolean inverse monoids

TL;DR

This paper develops a general theory of invariant means on Boolean inverse monoids, extending classical results related to paradoxical decompositions and etale groupoids through non-commutative Stone duality.

Contribution

It introduces the theory of invariant means on Boolean inverse monoids and characterizes when such monoids admit invariant means, generalizing classical Tarski-type results.

Findings

Characterization of Boolean inverse monoids admitting invariant means

Extension of classical paradoxical decomposition theory

Connection to etale topological groupoids via non-commutative Stone duality

Abstract

The classical theory of invariant means, which plays an important role in the theory of paradoxical decompositions, is based upon what are usually termed `pseudogroups'. Such pseudogroups are in fact concrete examples of the Boolean inverse monoids which give rise to etale topological groupoids under non-commutative Stone duality. We accordingly initiate the theory of invariant means on arbitrary Boolean inverse monoids. Our main theorem is a characterization of when a Boolean inverse monoid admits an invariant mean. This generalizes the classical Tarski alternative proved, for example, by de la Harpe and Skandalis, but using different methods.