|Mar. 28 - Apr. 04||RIMS 420||KTGU-IMU Mathematics Colloquia and Seminars|
|May 17|| 14:00 - 15:30
| Hiroshi Ando (Chiba)
Unitarizability, Maurey-Nikishin 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 $\ell^2$)
(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 a unitarily representable SIN Polish group which is not of finite type, answering the above question. Our analysis is based on 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.
|May 30 - June 03||RIMS 420||Geometric Analysis on Discrete Groups|
|Sep. 12-14||RIMS 420||Recent developments in operator algebras (program)|