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

Aldo Garcia Guinto; Yuki Miyamoto

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arXiv · 2510.26108

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

Aldo Garcia Guinto·Yuki Miyamoto

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Abstract

Given a locally compact second countable group $G$ with a 2-cocycle $ω$, we show that the restriction of the twisted Plancherel weight $φ^ω_G$ 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 $L_ω(G)$.

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BibTeX RIS

Aldo Garcia Guinto, Yuki Miyamoto. 2025-10-30. On modular invariants of twisted group von Neumann algebras of almost unimodular groups. https://arxiv.org/abs/2510.26108

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