On Polish Groups of Finite Type

Abstract

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 ${\mathcal{U}}_{\text{fin}}$. 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.