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THE ÉTALE GROUPOID OF AN INVERSE SEMIGROUP AS A GROUPOID OF FILTERS

Part of: Semigroups Selfadjoint operator algebras Special categories

Published online by Cambridge University Press:  15 May 2013

M. V. LAWSON ,
S. W. MARGOLIS  and
B. STEINBERG
M. V. LAWSON*
Affiliation:
Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, UK
S. W. MARGOLIS
Affiliation:
Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel email margolis@math.biu.ac.il
B. STEINBERG
Affiliation:
Department of Mathematics, The City College of New York, NAC 8/133, Convent Ave. at 138th Street, New York, NY 10031, USA email bsteinberg@ccny.cuny.edu
*
✉markl@ma.hw.ac.uk
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Abstract

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Paterson showed how to construct an étale groupoid from an inverse semigroup using ideas from functional analysis. This construction was later simplified by Lenz. We show that Lenz’s construction can itself be further simplified by using filters: the topological groupoid associated with an inverse semigroup is precisely a groupoid of filters. In addition, idempotent filters are closed inverse subsemigroups and so determine transitive representations by means of partial bijections. This connection between filters and representations by partial bijections is exploited to show how linear representations of inverse semigroups can be constructed from the groups occurring in the associated topological groupoid.

Keywords

Étale groupoidsinverse semigroupsfilters

MSC classification

Secondary: 20M18: Inverse semigroups 20M30: Representation of semigroups; actions of semigroups on sets 18B40: Groupoids, semigroupoids, semigroups, groups (viewed as categories) 46L05: General theory of $C^*$-algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 94 , Issue 2 , April 2013 , pp. 234 - 256
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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