Factoriality of twisted locally compact group von Neumann algebras

TL;DR

This paper constructs an example of a locally compact group with a Borel 2-cocycle where the untwisted von Neumann algebra is a factor, but the twisted version has a diffuse center, illustrating a novel phenomenon.

Contribution

It provides the first known example of a group where twisting by a 2-cocycle changes the von Neumann algebra's factoriality.

Findings

Untwisted algebra is a factor.

Twisted algebra has a diffuse center.

Demonstrates the impact of cocycles on algebra structure.

Abstract

In this short note, we construct an exotic example of a locally compact group $G$ with a Borel $2$-cocycle $\omega$ such that the non-twisted group von Neumann algebra $L(G)$ is a factor, while the twisted group von Neumann algebra $L_\omega(G)$ has a diffuse center.