## Departmental PhD Thesis Exam – Barry Rowe

Everyone welcome.  Refreshments will be served in the Math
Lounge before the exam.

DEPARTMENTAL PHD THESIS EXAM

Monday, April 4, 2011, 4:10 - 5:00 p.m.,
in BA 6183, 40 St. George Street

PhD Candidate:  Barry Rowe

Thesis Title:  The Left Regular Representation of a Semigroup
http://individual.utoronto.ca/browe/ut-thesis.pdf

Thesis Abstract:

As with groups, one can study the left regular representation L(S) of a
semigroup S. Specifically, I will consider the Hilbert space given by
l_2(S), and the weak operator topology closure of the algebra generated
by the operators L_s, where L_s acts as left translation by s on the
basis S.  If one considers such representations, then it is natural to
ask similar questions to the well known case where S is a group.

I start by formulating several questions in the semigroup case and then
work toward understanding the structure of the representations given.
Most of this concerns the problem of describing what the images of the
elements in the semigroup can look like in this representation
(specifically, what operators can arise as an L_s from some s in a
semigroup S).  Some additional results concerning the problem of
determining when the algebra L(S) is reflexive (that is, when L(S)
contains all the operators whose lattice of invariant subspaces
contains the lattice of invariant subspaces of L(S)), and results
concerning a universal algebra of such representations are also given.