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Amenable quantum groups are not always unitarizable. October 06, 2017 Seminars

Dr. Michael Brannan

Assistant Professor

Texas A&M University

One of the most elementary and important results in the representation theory of groups is that any representation of a finite group G as bounded operators on a Hilbert space can be ``averaged'' to a unitary representation. In fact, it was shown by Day and Dixmier in 1950 that the same result holds if G is assumed to be an amenable locally compact group and if our representation is assumed continuous and uniformly bounded. In short, ``amenable groups are unitarizable''. In this talk, I will give a light introduction to the concept of a locally compact quantum group (LCQG), which is an operator algebraic object that encodes and generalizes the structure of a locally compact group. After introducing the corresponding notions of amenability, uniformly bounded representations, and unitary/unitarizable representations of LCQGs, I will describe some joint work with Sang-Gyun Youn (Seoul National University) where we show that the expected quantum analogue the Day-Dixmier unitarizability theorem fails. Namely, there exist natural examples of amenable quantum groups that fail to be unitarizable.

Friday, October 20th

FLN 4.01.20

2:30 - 3:30 PM