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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 PhD Advisor: Peter Rosenthal 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.

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