Non-commutative Stone duality: inverse semigroups, topological groupoids and C*-algebras

Abstract

We study a non-commutative generalization of Stone duality that connects a class of inverse semigroups, called Boolean inverse $\wedge$-semigroups, with a class of topological groupoids, called Hausdorff Boolean groupoids. Much of the paper is given over to showing that Boolean inverse $\wedge$-semigroups arise as completions of inverse semigroups we call pre-Boolean. An inverse $\wedge$-semigroup is pre-Boolean if and only if every tight filter is an ultrafilter, where the definition of a tight filter is obtained by combining work of both Exel and Lenz. A simple necessary condition for a semigroup to be pre-Boolean is derived and a variety of examples of inverse semigroups are shown to satisfy it. Thus the polycyclic inverse monoids, and certain Rees matrix semigroups over the polycyclics, are pre-Boolean and it is proved that the groups of units of their completions are precisely the Thompson-Higman groups $G_{n,r}$. The inverse semigroups arising from suitable directed graphs are also pre-Boolean and the topological groupoids arising from these graph inverse semigroups under our non-commutative Stone duality are the groupoids that arise from the Cuntz-Krieger $C^{\ast }$-algebras.

The presentation has been sharpened up and some minor errors corrected

References in corpus (3)

• The Local Structure of Tilings and their Integer Group of Coinvariants
• Topological equivalence of tilings
• Trees, ultrametrics, and noncommutative geometry

Cited by in corpus (2)

• Pseudogroups and their etale groupoids
• The structure of generalized inverse semigroups