A duality principle for groups
Abbreviated Journal Title
J. Funct. Anal.
Group representations; Frame vectors; Bessel vectors; Duality principle; Von Neumann algebras; II(1) factors; WEYL-HEISENBERG FRAMES; GABOR FRAMES; II1 FACTORS; REPRESENTATIONS; Mathematics
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 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. (C) 2009 Elsevier Inc. All rights reserved.
Journal of Functional Analysis
"A duality principle for groups" (2009). Faculty Bibliography 2000s. 1496.