Projective representations of almost unimodular groups

Given an almost unimodular $G$, so that the Plancherel weight $\varphi_G$ on the group von Neumann algebra $L(G)$ is almost periodic, we show that the basic construction for the inclusion $L(G)^{\varphi_G} \leq L(G)$ is isomorphic to a twisted group von Neumann algebra of $G \times \Delta_G(G)\hat{\ }$ with a continuous 2-cocycle, where $\Delta_G$ 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 $\mathbb{T} \rtimes_{(1,\omega)} G$ is almost unimodular. Using this result and the connection between $\omega$-projective representations of $G$ and the representations of $\mathbb{T} \rtimes_{(1,\omega)} G$, 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.