Journal of the Australian Mathematical Society

Research Article

A NONCOMMUTATIVE GENERALIZATION OF STONE DUALITY

M. V. Lawsona1

a1 Department of Mathematics and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, Scotland (email: M.V.Lawson@ma.hw.ac.uk)

Abstract

We prove that the category of boolean inverse monoids is dually equivalent to the category of boolean groupoids. This generalizes the classical Stone duality between boolean algebras and boolean spaces. As an instance of this duality, we show that the boolean inverse monoid Cn associated with the Cuntz groupoid Gn is the strong orthogonal completion of the polycyclic (or Cuntz) monoid Pn. The group of units of Cn is the Thompson group Vn,1.

(Received November 15 2009)

(Accepted January 26 2010)

2000 Mathematics subject classification

  • primary 20M18; secondary 18B40;
  • 06E15

Keywords and phrases

  • inverse semigroup;
  • étale topological groupoid;
  • stone duality

Footnotes

This research was supported by an EPSRC grant (EP/F004184, EP/F014945, EP/F005881), the Fundação para a Ciência e a Tecnologia, courtesy of Pedro Resende under the grant PPCDT/MAT/55958/2004, Groupoids and quantales in geometry and analysis, at the Instituto Superior Técnico, Lisbon, and by Prof. Stuart Margolis of Bar-Ilan University, Israel.