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Inverse Semigroups, Groupoids and A Problem of J. Renault

In: Algebraic Methods in Operator Theory

Author

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  • Alan L. T. Paterson

    (University of Mississippi, Department of Mathematics)

Abstract

In [8], J. Renault showed that a topological groupoid G relates to inverse semigroups through its ample semigroup G a.In the case when G is r-discrete, the semigroup G a is “large” and determines the topology of G. The main theme of this paper is the following natural problem which was raised by Renault: given an inverse semigroup S (assumed, for convenience, unital) does there exist a (Hausdorff) r-discrete groupoid G with S “determining” G as an inverse subsemigroup of G a? What kind of uniqueness can we expect? Renault shows that there exists such a groupoid in the case where S is the Cuntz inverse semigroup O n, the groupoid G in that case being the Cuntz groupoid O n. We show that the answer to both questions is negative in general. The natural class of groupoids associated with an inverse semigroup S is that of S-groupoids. Such groupoids are not always Hausdorff but there always exists a faithful S-groupoid whose representation theory is essentially the same as that of S We describe briefly a construction which produces this and many other S-groupoids.

Suggested Citation

  • Alan L. T. Paterson, 1994. "Inverse Semigroups, Groupoids and A Problem of J. Renault," Springer Books, in: Raúl E. Curto & Palle E. T. Jørgensen (ed.), Algebraic Methods in Operator Theory, pages 79-89, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-0255-4_10
    DOI: 10.1007/978-1-4612-0255-4_10
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