Speaker:  Alan Paterson, University of Missippi

Title:  Inverse semigroups, groupoids and their operator algebras

Abstract: Inverse semigroups can be thought of as semigroups of partial
one-to-one maps which are closed under inversion, or equivalently, as
*-semigroups of partial isometries on a Hilbert space.  A groupoid is
roughly a set with a partially defined multiplication such that the
usual group axioms hold whenever they make sense.  (Precisely, a groupoid is a
small category with inverses.)  At first sight, there does not seem to
be much in common between inverse semigroups, groupoids and operator
algebras.  But in fact there are close connections between them, and the
lecture will informally discuss some of these. We will describe how
operator algebras arise from the representation theories of both inverse
semigroups and locally compact groupoids, and then talk about the
construction of a canonical groupoid for every inverse semigroup.  This
groupoid is called the universal groupoid and the elements of the
inverse semigroup are realized as subsets of the groupoid.  The representation
theories of the inverse semigroup and the groupoid are the same, and the
well-developed representation theory of groupoids can thus be used to
investigate the representation theory of inverse semigroups.



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