A Parametrization of the Irreducible Representations of a Compact Inverse Semigroup

Abstract

The aim in this article is to provide a parametrization of the finite dimensional irreducible representations of a compact inverse semigroup in terms of the irreducible representations of maximal subgroups and order theoretic properties of the idempotent set. As a consequence, we obtain a new, and more conceptual, proof of the following theorem of Shneperman: a compact inverse semigroup has enough finite dimensional irreducible representations to separate points if and only if its idempotent set is totally disconnected. Moreover, we also prove that every norm continuous irreducible *-representation of a compact inverse semigroup on a Hilbert space is finite dimensional.

Key Words:

• Compact inverse semigroups
• Representation theory
• Semilattice

2010 Mathematics Subject Classification:

• 20M10

ACKNOWLEDGMENTS

This work constitutes part of the Ph.D. thesis of the first author, written at the University of Ottawa under the supervision of David Handelman and Benjamin Steinberg.

Notes

Communicated by V. Gould.