May 17 
14:00  15:30 RIMS 206 
Hiroshi Ando (Chiba)
Unitarizability, MaureyNikishin factorization and Polish groups of finite type
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 II$_1$ factor 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 $\ell^2$)
and
(b) $G$ is SIN, i.e., $G$ admits a twosided 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 a unitarily representable SIN Polish group which is not of finite type, answering the above question. Our analysis is based on the MaureyNikishin 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.
