On modular invariants of twisted group von Neumann algebras of almost unimodular groups

TL;DR

This paper investigates the structure and invariants of twisted group von Neumann algebras for almost unimodular groups, revealing conditions for semifiniteness, factorization, and modular spectrum characterization.

Contribution

It establishes the semifiniteness criterion for subalgebras, represents the algebra as a cocycle action, and characterizes the modular spectrum for these algebras.

Findings

Semifiniteness of subalgebras is equivalent to the subgroup being open.

Representation of the algebra as a cocycle action of the modular function group.

Characterization of when the algebra is a factor and its modular spectrum.

Abstract

Given a locally compact second countable group $G$ with a 2-cocycle $\omega$, we show that the restriction of the twisted Plancherel weight $\varphi^\omega_G$ to the subalgebra generated by a closed subgroup $H$ in the twisted group von Neumann algebra $L_\omega(G)$ is semifinite if and only if $H$ is open. When $G$ is almost unimodular, i.e. $\ker\Delta_G$ is open, we show that $L_\omega(G)$ can be represented as a cocycle action of the $\Delta_G(G)$ on $L_\omega(\ker\Delta_G)$ and the basic construction of the inclusion $L_\omega(\ker\Delta_G)\leq L_\omega(G)$ can be realized as a twisted group von Neumann algebra of $\Delta_G(G)\hat{\ } \times G$, where $\Delta_G$ is the modular function. Furthermore, when $G$ has a sufficiently large non-unimodular part, we give a characterization of $L_\omega(G)$ being a factor and provide a formula for the modular spectrum of $L_\omega(G)$.