The density theorem for projective representations via twisted group von Neumann algebras

We consider converses to the density theorem for irreducible, projective, unitary group representations restricted to lattices using the dimension theory of Hilbert modules over twisted group von Neumann algebras. We show that under the right assumptions, the restriction of a $\sigma$-projective unitary representation $\pi$ of a group $G$ to a lattice $\Gamma$ extends to a Hilbert module over the twisted group von Neumann algebra $\text{L}(\Gamma,\sigma)$. We then compute the center-valued von Neumann dimension of this Hilbert module. For abelian groups with 2-cocycle satisfying Kleppner's condition, we show that the center-valued von Neumann dimension reduces to the scalar value $d_{\pi} \text{vol}(G/\Gamma)$, where $d_{\pi}$ is the formal dimension of $\pi$ and $\text{vol}(G/\Gamma)$ is the covolume of $\Gamma$ in $G$. We apply our results to characterize the existence of multiwindow super frames and Riesz sequences associated to $\pi$ and $\Gamma$. In particular, we characterize when a lattice in the time-frequency plane of a second countable, locally compact abelian group admits a Gabor frame or Gabor Riesz sequence.