An inner amenable group whose von Neumann algebra does not have property Gamma

TL;DR

This paper constructs a specific inner amenable group with infinite conjugacy classes whose associated von Neumann algebra lacks property Gamma, addressing a long-standing open problem in operator algebras.

Contribution

It provides the first example of an inner amenable group with these properties, solving Effros's 1975 problem.

Findings

Constructed an inner amenable group with infinite conjugacy classes

Associated von Neumann algebra does not have property Gamma

Addresses a problem posed by Effros in 1975

Abstract

We construct inner amenable groups G with infinite conjugacy classes and such that the associated II_1 factor does not have property Gamma of Murray and von Neumann. This solves a problem posed by Effros in 1975.