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Positive Definite Functions and Representation Theory of Locally Compact Groups

Emil Axelgaard
Fredag, 23 april, 2010, at 14:15-15:15, in Aud. D3 (1531-215)
A locally compact group is a topological group whose topology is locally compact and Hausdorff. Such an object can be analyzed by studying its unitary representations which are continuous (in a certain topology) group homomorphisms into the group of unitary operators on a Hilbert space.

One way to construct such representations is by means of positive definite functions and the Gelfand-Naimark-Segal construction. An obvious question is of course whether representations arising in this way are irreducible. It turns out that the answer reveals an intimate and exciting relation between representation theory, convexity theory and the theory of functional equations.

In this talk, I will outline how positive definite functions give rise to unitary representations, and explain how the question of irreducibility of such representations can be studied through the use of convexity theory. By focusing on a smaller class of unitary representations, I will then make it clear how a certain functional equation suddenly becomes the central object of attention. Along the way, concepts such as Gelfand pairs, spherical functions and the Bochner-Godement theorem will appear. If time permits, I will explain how these ideas generalize to inductive limits of locally compact groups.

I will try to give careful definitions and thorough explanations of all concepts and will not assume knowledge of any material which is not covered by the course in advanced analysis. Hence, the seminar should be accessible to anyone with a basic knowledge of abstract functional analysis.

Kontaktperson: Magnus Roed Lauridsen