Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T14:58:08.019Z Has data issue: false hasContentIssue false
  • English
  • Français

A NONCOMMUTATIVE GENERALIZATION OF STONE DUALITY

Part of: Semigroups

Published online by Cambridge University Press:  05 May 2010

M. V. Lawson
M. V. Lawson*
Affiliation:
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)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

inverse semigroupétale topological groupoidstone duality

MSC classification

Secondary: 20M18: Inverse semigroups 06E15: Stone spaces (Boolean spaces) and related structures
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 88 , Issue 3 , June 2010 , pp. 385 - 404
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Bratteli, O. and Jorgensen, P. E. T., ‘Iterated function systems and permutation representations of the Cuntz algebra’, Mem. Amer. Math. Soc. 139 (1999), 663.Google Scholar
[2]Burris, S. and Sankappanavar, H. P., A Course in Universal Algebra, millennium edition, freely available at www.math.uwaterloo.ca/∼snburris/htdocs.html.CrossRefGoogle Scholar
[3]Cuntz, J., ‘Simple C *-algebras generated by isometries’, Comm. Math. Phys. 57 (1977), 173–185.CrossRefGoogle Scholar
[4]Exel, R., ‘Inverse semigroups and combinatorial C *-algebras’, Bull. Braz. Math. Soc. (N.S.) 39 (2008), 191–313.CrossRefGoogle Scholar
[5]Exel, R., ‘Tight representations of semilattices and inverse semigroups’, Semigroup Forum 79 (2009), 159–182.CrossRefGoogle Scholar
[6]Exel, R., ‘Reconstructing a totally disconnected groupoid from its ample semigroup’, Proc. Amer. Math. Soc., to appear. arXiv:0907.4530v1.Google Scholar
[7]Higgins, Ph. J., Categories and Groupoids (Van Nostrand Reinhold, London, 1971).Google Scholar
[8]Hughes, B., ‘Trees and ultrametric spaces: a categorical equivalence’, Adv. Math. 189 (2004), 148–191.CrossRefGoogle Scholar
[9]Hughes, B., ‘Trees, ultrametrics and noncommutative geometry’, Pure Appl. Math. Q., to appear. arXiv:math/0605131v2.Google Scholar
[10]Johnstone, P. T., Stone Spaces (Cambridge University Press, Cambridge, 1986).Google Scholar
[11]Kawamura, K., ‘Polynomial endomorphisms of the Cuntz algebras arising from permutations I. General theory’, Lett. Math. Phys. 71 (2005), 149–158.CrossRefGoogle Scholar
[12]Kawamura, K., ‘Branching laws for polynomial endomorphisms of the Cuntz algebras arising from permutations’, Lett. Math. Phys. 77 (2006), 111–126.CrossRefGoogle Scholar
[13]Kellendonk, J., ‘The local structure of tilings and their integer group of coinvariants’, Commun. Math. Phys. Soc. 187 (1997), 115–157.CrossRefGoogle Scholar
[14]Kellendonk, J., ‘Topological equivalence of tilings’, J. Math. Phys. 38 (1997), 1823–1842.CrossRefGoogle Scholar
[15]Lawson, M. V., ‘Coverings and embeddings of inverse semigroups’, Proc. Edinb. Math. Soc. 36 (1996), 399–419.CrossRefGoogle Scholar
[16]Lawson, M. V., Inverse Semigroups (World Scientific, Singapore, 1998).CrossRefGoogle Scholar
[17]Lawson, M. V., ‘Orthogonal completions of the polycyclic monoids’, Comm. Algebra 35 (2007), 1651–1660.CrossRefGoogle Scholar
[18]Lawson, M. V., ‘The polycyclic monoids P n and the Thompson groups V n,1’, Comm. Algebra 35 (2007), 4068–4087.CrossRefGoogle Scholar
[19]Lawson, M. V., ‘Primitive partial permutation representations of the polycyclic monoids and branching function systems’, Period. Math. Hungar. 58 (2009), 189–207.CrossRefGoogle Scholar
[20]Lawson, M. V., Margolis, S. and Steinberg, B., ‘Inverse semigroup actions and a class of étale groupoids’, in preparation.Google Scholar
[21]Leech, J., ‘Inverse monoids with a natural semilattice ordering’, Proc. Lond. Math. Soc. (3) 70 (1995), 146–182.CrossRefGoogle Scholar
[22]Lenz, D. H., ‘On an order-based construction of a topological groupoid from an inverse semigroup’, Proc. Edinb. Math. Soc. 51 (2008), 387–406.CrossRefGoogle Scholar
[23]Nivat, M. and Perrot, J.-F., ‘Une généralisation du monoïde bicyclique’, C. R. Acad. Sci. Paris 271 (1970), 824–827.Google Scholar
[24]Paterson, A. L. T., Groupoids, Inverse Semigroups, and their Operator Algebras (Birkhäuser, Boston, MA, 1999).CrossRefGoogle Scholar
[25]Renault, J., A Groupoid Approach to C *-Algebras (Springer, Berlin, 1980).CrossRefGoogle Scholar
[26]Resende, P., ‘Lectures on étale groupoids, inverse semigroups and quantales’, August 2006.Google Scholar
[27]Resende, P., ‘Étale groupoids and their quantales’, Adv. Math. 208 (2007), 147–209.CrossRefGoogle Scholar
[28]Steinberg, B., ‘A groupoid approach to discrete inverse semigroup algebras’, arXiv:0903.3456v1.Google Scholar
[29]Stone, M. H., ‘Applications of the theory of Boolean rings to general topology’, Trans. Amer. Math. Soc. 41 (1937), 375–481.CrossRefGoogle Scholar