Square Integrable Representations, von Neumann Algebras and An Application to Gabor Analysis

Abstract

Let G be a locally compact group, and let Γ be a lattice in G. Let (π, Hπ ) be an irreducible, square integrable unitary representation of G. Then the restriction of π to Γ extends to VN(Γ), the von Neumann algebra generated by the regular representation of Γ. We determine the center valued (or extended) von Neumann dimension of Hπ as a VN(Γ)-module. This result is extended to representations which are square integrable modulo the center of G. An explicit formula is given when G is a semisimple algebraic group and when G is a nilpotent Lie group. As an application, we study the question of the existence of frames for the discretized windowed Fourier transform.