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On Polish Groups of Finite Type
Hiroshi Ando (1) and Yasumichi Matsuzawa (2)
(1) Research Institute for Mathematical Sciences, Kyoto University, 606-8502, KYOTO, JAPAN(2) Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-ku, 060-0810, SAPPORO, JAPAN
Sorin Popa initiated the study of Polish groups which are embeddable into the unitary group of a separable finite von Neumann algebra. Such groups are called of finite type or said to belong to the class $\mathscr{U}_{\text{fin}}$. We give necessary and sufficient conditions for Polish groups to be of finite type, and construct examples of such groups from I$_{\infty}$ and II$_{\infty}$ von Neumann algebras. We also discuss permanence properties of finite type groups under various algebraic operations. Finally we close the paper with some questions concerning Polish groups of finite type.
Keywords: bi-invariant metric, class $\mathscr{U}_{\text{fin}}$, finite type group, Polish group, positive definite function, SIN-group, II$_1$ factor