Similarity problem and fixed point theorems

04/10/2019, 14:00 - 15:00 in MAP/0G/017

Turowska, Lyudmila (Chalmers University Gothenburg)


Abstract:

Let G be a group and π:G→ℒ(H) be its representation in the space of bounded linear operators on a Hilbert space. One of the general problems which motivated the results of this talk is to find conditions under which π is unitarizable, i.e. similar to a representation ρ such that ρ(g) is unitary for all gG. An obvious necessary condition is the boundedness of π: supgG||π(g)||<∞. Day and Dixmier proved that this condition is sufficient if the group G is amenable. However there are non-amenable groups with non-unitarizable bounded representations. Note that it is a challenging open problem to identify all groups whose all representations are unitarizable. We look for conditions on representations (not on groups) under which they are similar to unitary ones. Our result is inspired by the theory of operators on spaces with indefinite metric. We show that a bounded representation π is unitarizable if it preserves a quadratic form with finite number of negative squares. Our approach is based on connection between such representations and biholomorphic transformations of the operator ball. The existence of fixed points for groups of biholomorphic transformations is the key result. The talk is based on a joint work with M. Ostrovskii and V. Shulman from 2011 and some recent developments.

Mathematical Sciences Research Centre