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

Abstract

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 σ-projective unitary representation π of a group G to a lattice Γ extends to a Hilbert module over the twisted group von Neumann algebra L(Γ, σ). 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 dπ vol(G/Γ), where dπ is the formal dimension of π and vol(G/Γ) is the covolume of Γ in G. We apply our results to characterize the existence of multiwindow super frames and Riesz sequences associated to π and Γ. 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.

Smooth lattice orbits of nilpotent groups and strict comparison of projections

“…By Proposition 3.5, this representation can be extended to give H π the structure of a Hilbert L(Γ, σ)-module, where the action is determined by λ σ Γ (γ) • f = π(γ)f for γ ∈ Γ and f ∈ H π . The dimension dim (L(Γ,σ),τ ) H π of this Hilbert L(Γ, σ)-module was computed in [34] to be vol(G/Γ)d π , see [34,Theorem 4.3]. Therefore, by Proposition 5.3, it follows that τ (E) = dim (L(Γ,σ),τ ) H π = vol(G/Γ)d π , as required.…”

“…Theorem 6.6 can be extended to multi-window and super systems, cf. [34] for these notions. Under Kleppner's condition, the inequality vol(G/Γ)d π < n/d (resp.…”

“…Representations of this type can be treated as projective representations of G = N/Z, so-called projective relative discrete series representations (see Section 6.1). The following result can be derived from [12,34] by combining [34,Theorem 1.3] and the arguments underlying [12,Theorem 3].…”

“…The question of the existence of a general vector g ∈ H π generating a frame π(Γ)g in H π (see Theorem 1.2) can be naturally approached using techniques for von Neumann algebras and their Hilbert modules, as shown by Bekka [12] (cf. [34] for Riesz sequences). The fundamental observation here is that there exists a frame (resp.…”

“…By the square-integrability of π, the restriction π| Γ can be extended to give H π the structure of a Hilbert L(Γ, σ)-module, so that the existence of frames (resp. Riesz sequences) in Proposition 1] and [34,Theorem 5.1]. When L(Γ, σ) is a factor, such submodule inclusions are in turn equivalent to inequalities involving the associated von Neumann dimensions, which gives rise to the inequalities vol(G/Γ)d π ≤ 1 (resp.…”