Projective representations of almost unimodular groups

Given an almost unimodular $G$, so that the Plancherel weight ${\phi }_G$ on the group von Neumann algebra $L(G)$ is almost periodic, we show that the basic construction for the inclusion $L(G{)}^{{\phi }_G}\le L(G)$ is isomorphic to a twisted group von Neumann algebra of $G\times {\mathrm{\Delta }}_G(G)\widehat{}$ with a continuous 2-cocycle, where ${\mathrm{\Delta }}_G$ is the modular function. We show that when $G$ is second countable and admits a Borel 2-cocycle, $G$ is almost unimodular if and only if the central extension $\mathbb{T}{⋊}_{(1,\omega )}G$ is almost unimodular. Using this result and the connection between $\omega$-projective representations of $G$ and the representations of $\mathbb{T}{⋊}_{(1,\omega )}G$, we show that the formal degrees of irreducible and factorial square integrable projective representations behaved similarly to their representations counterparts and obtain the Atiyah–Schmid formula in the setting of second countable almost unimodular groups with a 2-cocycle twist and a finite covolume subgroup, which uses the Murray–von Neumann dimension for certain Hilbert space modules over the twisted group von Neumann algebra with its twisted Plancherel weight.