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 Un . We give necessary and sufficient conditions for Polish groups to be of finite type, and construct exmaples of such groups from I1 and II1 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.