W. D. Munn proved that a finite dimensional representation of an inverse semigroup
is equivalent to a ⋆-representation if and only if it is bounded. The first goal of this
thesis will be to give new analytic proof that every finite dimensional representation
of a compact inverse semigroup is equivalent to a ⋆-representation.
The second goal is to parameterize all finite dimensional irreducible representations
of a compact inverse semigroup in terms of maximal subgroups and order
theoretic properties of the idempotent set. As a consequence, we obtain a new and
simpler 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.
Our last theorem is the following: every norm continuous irreducible ∗-representation
of a compact inverse semigroup on a Hilbert space is finite dimensional.
| Identifer | oai:union.ndltd.org:uottawa.ca/oai:ruor.uottawa.ca:10393/20183 |
| Date | January 2011 |
| Creators | Hajji, Wadii |
| Contributors | Handelman David, Steinberg, Benjamin |
| Publisher | Université d'Ottawa / University of Ottawa |
| Source Sets | Université d’Ottawa |
| Language | English |
| Detected Language | English |
| Type | Thesis |
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