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

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 $L_\omega(G)$.