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A NONCOMMUTATIVE GENERALIZATION OF STONE DUALITY

  • M. V. Lawson (a1)
Abstract
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.

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[3]J. Cuntz , ‘Simple C*-algebras generated by isometries’, Comm. Math. Phys. 57 (1977), 173185.

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[13]J. Kellendonk , ‘The local structure of tilings and their integer group of coinvariants’, Commun. Math. Phys. Soc. 187 (1997), 115157.

[14]J. Kellendonk , ‘Topological equivalence of tilings’, J. Math. Phys. 38 (1997), 18231842.

[16]M. V. Lawson , Inverse Semigroups (World Scientific, Singapore, 1998).

[17]M. V. Lawson , ‘Orthogonal completions of the polycyclic monoids’, Comm. Algebra 35 (2007), 16511660.

[18]M. V. Lawson , ‘The polycyclic monoids Pn and the Thompson groups Vn,1’, Comm. Algebra 35 (2007), 40684087.

[19]M. V. Lawson , ‘Primitive partial permutation representations of the polycyclic monoids and branching function systems’, Period. Math. Hungar. 58 (2009), 189207.

[24]A. L. T. Paterson , Groupoids, Inverse Semigroups, and their Operator Algebras (Birkhäuser, Boston, MA, 1999).

[25]J. Renault , A Groupoid Approach to C*-Algebras (Springer, Berlin, 1980).

[27]P. Resende , ‘Étale groupoids and their quantales’, Adv. Math. 208 (2007), 147209.

[29]M. H. Stone , ‘Applications of the theory of Boolean rings to general topology’, Trans. Amer. Math. Soc. 41 (1937), 375481.

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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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