Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bessel duality of Gabor systems: A von Neumann algebraic perspective

Ulrik Enstad, Franz Luef

TL;DR

The paper recasts Bessel duality for Gabor systems within a von Neumann algebra bimodule framework. It proves a general result: for a bifinite $M$-$N$ bimodule with aligned traces and coinciding centers, left and right bounded vectors coincide, with a quantitative norm bound. Applying this to the Gabor bimodule $L^2(G)$ over the twisted group von Neumann algebras of a lattice and its adjoint shows that Bessel duality emerges naturally from the operator-algebraic structure, even beyond factorial settings. This provides an operator-algebraic interpretation of Gabor duality and connects well-localized time-frequency analysis to Morita-type frameworks in twisted group algebras.

Abstract

Bessel duality of regular Gabor systems states that a Gabor system over a lattice is a Bessel sequence if and only if the corresponding Gabor system over the adjoint lattice is a Bessel sequence. We show that this fundamental result of time-frequency analysis can be deduced from a theorem in the theory of bimodules over von Neumann algebras, namely that under certain conditions, their left and right bounded vectors coincide.

Bessel duality of Gabor systems: A von Neumann algebraic perspective

TL;DR

The paper recasts Bessel duality for Gabor systems within a von Neumann algebra bimodule framework. It proves a general result: for a bifinite - bimodule with aligned traces and coinciding centers, left and right bounded vectors coincide, with a quantitative norm bound. Applying this to the Gabor bimodule over the twisted group von Neumann algebras of a lattice and its adjoint shows that Bessel duality emerges naturally from the operator-algebraic structure, even beyond factorial settings. This provides an operator-algebraic interpretation of Gabor duality and connects well-localized time-frequency analysis to Morita-type frameworks in twisted group algebras.

Abstract

Bessel duality of regular Gabor systems states that a Gabor system over a lattice is a Bessel sequence if and only if the corresponding Gabor system over the adjoint lattice is a Bessel sequence. We show that this fundamental result of time-frequency analysis can be deduced from a theorem in the theory of bimodules over von Neumann algebras, namely that under certain conditions, their left and right bounded vectors coincide.

Paper Structure

This paper contains 8 sections, 10 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Bimodules over nonfactors
  3. Modules
  4. Center-valued von Neumann dimension
  5. Finitely generated modules
  6. Restriction to a subalgebra
  7. Bounded vectors
  8. Gabor systems

Key Result

theorem 1.1

For a lattice $\Delta$ in $G \times \widehat{G}$ with adjoint lattice $\Delta^{\circ}$ (see eq:adjoint-lattice) and $g \in L^2(G)$, the following are equivalent:

Theorems & Definitions (22)

  • theorem 1.1: Bessel duality
  • theorem 1.2
  • proposition 2.1
  • proof
  • lemma 2.2
  • proof
  • proposition 2.3
  • proof
  • definition 2.4
  • lemma 2.5
  • ...and 12 more