A duality principle for groups

    Authors

    D. Dutkay; D. G. Han;D. Larson

    Comments

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    Abbreviated Journal Title

    J. Funct. Anal.

    Keywords

    Group representations; Frame vectors; Bessel vectors; Duality principle; Von Neumann algebras; II(1) factors; WEYL-HEISENBERG FRAMES; GABOR FRAMES; II1 FACTORS; REPRESENTATIONS; Mathematics

    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 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 Title

    Journal of Functional Analysis

    Volume

    257

    Issue/Number

    4

    Publication Date

    1-1-2009

    Document Type

    Article

    Language

    English

    First Page

    1133

    Last Page

    1143

    WOS Identifier

    WOS:000267296600008

    ISSN

    0022-1236

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