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 to lattices using the dimension theory of Hilbert modules over twisted group von Neumann algebras. We show that the restriction of such a sigma-projective unitary representation pi of a unimodular, second-countable group G to a lattice Gamma extends to a Hilbert module over the twisted group von Neumann algebra of (Gamma, sigma). We then compute the center-valued von Neumann dimension of this Hilbert module. For abelian groups with 2-co cycle satisfying Kleppner's condition, we show that the center-valued von Neumann dimension reduces to the scalar value d pi vol(G/Gamma), where d pi is the formal dimension of pi and 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. (c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).