On modular invariants of twisted group von Neumann algebras of almost unimodular groups
Aldo Garcia Guinto, Yuki Miyamoto
TL;DR
This work analyzes twisted group von Neumann algebras $L_ω(G)$ for locally compact groups with a 2-cocycle $ω$, focusing on modular invariants and semifiniteness. The authors prove that the twisted Plancherel weight $φ^ω_G$ is semifinite on the subalgebra generated by a closed subgroup $H$ if and only if $H$ is open, extending previous results to the twisted setting. When $G$ is almost unimodular, they decompose $L_ω(G)$ as a cocycle crossed product $L_ω(ker Δ_G) ⋊_{(\check{α}^ω,u^ω)} Δ_G(G)$ and show the basic construction corresponds to a twisted group von Neumann algebra on $Δ_G(G)^\hat × G$, with a dual action given by the point modular extension. They also characterize factoriality and compute the modular spectrum $S(L_ω(G))$ via the action on the center of $L_ω(ker Δ_G)$, and provide criteria for semifiniteness in terms of a $T$-invariant, yielding examples of type III factors. These results unify and extend prior discrete and unimodular cases to the twisted, almost unimodular setting, offering tools to identify when twisted algebras are factors and to determine their spectral invariants.
Abstract
Given a locally compact second countable group $G$ with a 2-cocycle $ω$, we show that the restriction of the twisted Plancherel weight $\varphi^ω_G$ to the subalgebra generated by a closed subgroup $H$ in the twisted group von Neumann algebra $L_ω(G)$ is semifinite if and only if $H$ is open. When $G$ is almost unimodular, i.e. $\kerΔ_G$ is open, we show that $L_ω(G)$ can be represented as a cocycle action of the $Δ_G(G)$ on $L_ω(\kerΔ_G)$ and the basic construction of the inclusion $L_ω(\kerΔ_G)\leq L_ω(G)$ can be realized as a twisted group von Neumann algebra of $Δ_G(G)\hat{\ } \times G$, where $Δ_G$ is the modular function. Furthermore, when $G$ has a sufficiently large non-unimodular part, we give a characterization of $L_ω(G)$ being a factor and provide a formula for the modular spectrum of $L_ω(G)$.