On Polish Groups of Finite Type

TL;DR

This paper characterizes Polish groups of finite type, providing necessary and sufficient conditions, constructing examples from semifinite von Neumann algebras, and exploring their algebraic properties.

Contribution

It offers a complete characterization of Polish groups of finite type and introduces new examples derived from semifinite von Neumann algebras.

Findings

Necessary and sufficient conditions for finite type groups

Construction of examples from semifinite von Neumann algebras

Analysis of permanence properties under algebraic operations

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