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JOINS AND COVERS IN INVERSE SEMIGROUPS AND TIGHT ${C}^{\ast } $-ALGEBRAS

Part of: Semigroups Selfadjoint operator algebras Groups and semigroups of linear operators, their generalizations and applications

Published online by Cambridge University Press:  10 January 2014

ALLAN P. DONSIG  and
DAVID MILAN
ALLAN P. DONSIG*
Affiliation:
Mathematics Department, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, USA
DAVID MILAN
Affiliation:
Department of Mathematics, University of Texas at Tyler, Tyler, TX 75799, USA email dmilan@uttyler.edu
*
✉adonsig1@unl.edu
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Abstract

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We show Exel’s tight representation of an inverse semigroup can be described in terms of joins and covers in the natural partial order. Using this, we show that the ${C}^{\ast } $-algebra of a finitely aligned category of paths, developed by Spielberg, is the tight ${C}^{\ast } $-algebra of a natural inverse semigroup. This includes as a special case finitely aligned higher-rank graphs: that is, for such a higher-rank graph $\Lambda $, the tight ${C}^{\ast } $-algebra of the inverse semigroup associated to $\Lambda $ is the same as the ${C}^{\ast } $-algebra of $\Lambda $.

Keywords

C∗-algebrainverse semigroupcategory of paths

MSC classification

Secondary: 46L05: General theory of $C^*$-algebras 20M18: Inverse semigroups 47D03: Groups and semigroups of linear operators
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 90 , Issue 1 , August 2014 , pp. 121 - 133
Copyright
Copyright ©2014 Australian Mathematical Publishing Association Inc. 

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