# Operator Algebra Seminar – University of Copenhagen

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# Operator Algebra Seminar

Speaker:  Hiroshi Ando (Chiba University)

Title: Unitarizability, Maurey-Nikishin factorization and Polish groups of finite type

Abstract: In the seminal work of cocylcle superrigidity theorem, Sorin Popa introduced the class of finite type Polish groups. A Polish group G is of finite type, if it is embeddable into the unitary group of a separable IIfactor equipped with the strong operator topology. Popa proposed a problem of finding abstract characterization of finite type Polish groups. As Popa pointed out, there are two conditions which are clearly necessary for a Polish group G to be of finite type, namely that

(a) G is unitarily representable (i.e., G is embeddable into the full unitary group of ℓ2

and

(b) G is SIN, i.e., G admits a two-sided invariant metric compatible with the topology.

Popa asked whether these two conditions are actually sufficient. In 2011, Yasumichi Matsuzawa and I obtained several partial positive answers for some classes of Polish groups.

In this talk, we show that there exists unitarily representable SIN Polish groups which are not of finite type, answering the above question. Along the way we find an unexpected connection between unitarizability of uniformly bounded representations of discrete groups and the study of finite type Polish groups.

Also, a key role is played by the Maurey-Nikishin Theorem on bounded maps from a Banach space of Rademacher type 2 to the space of all measurable maps on a probability space.

This is joint work with Yasumichi Matsuzawa, Andreas Thom, and Asger Törnquist. arXiv:1605.06909