A Duality principle for groups

TL;DR

This paper extends the duality principle from Gabor frames to dual pairs of projective unitary representations of countable groups, exploring existence conditions and connections with von Neumann algebra classification.

Contribution

It generalizes the duality principle to broader group representations and links the existence of dual pairs to classification problems in operator algebras.

Findings

Duality principle extends to projective unitary representations of countable groups.

Existence of dual pairs always for subrepresentations of abelian or amenable ICC groups.

Existence problem for free groups relates to classification of free group von Neumann algebras.

Abstract

The duality principle for Gabor frames states that a Gabor sequence obtained by a time-frequency lattice is a frame for $L^{2}(\R^{d})$ if and only if the associated adjoint Gabor sequence is a Riesz sequence. We prove that this duality principle extends to any dual pairs of projective unitary representations of countable groups. We examine the existence problem of dual pairs and establish some connection with classification problems for $\rm II_{1}$ factors. While in general such a pair may not exist for some groups, we show that such a dual pair always exists for every subrepresentation of the left regular unitary representation when $G$ is an abelian infinite countable group or an amenable ICC group. For free groups with finitely many generators, the existence problem of such a dual pair is equivalent to the well known problem about the classification of free group von Neumann algebras.