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

Given a locally compact second countable group $G$ with a 2-cocycle $\omega$, we show that the restriction of the twisted Plancherel weight ${\phi }_G^{\omega }$ 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. $\ker {\mathrm{\Delta }}_G$ is open, we show that $L_{\omega }(G)$ can be represented as a cocycle action of the ${\mathrm{\Delta }}_G(G)$ on $L_{\omega }(\ker {\mathrm{\Delta }}_G)$ and the basic construction of the inclusion $L_{\omega }(\ker {\mathrm{\Delta }}_G)\le L_{\omega }(G)$ can be realized as a twisted group von Neumann algebra of ${\mathrm{\Delta }}_G(G)\widehat{}\times G$, where ${\mathrm{\Delta }}_G$ 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 $L_{\omega }(G)$.