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

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