Dilations for systems of imprimitivity acting on Banach spaces

    Authors

    D. Han; D. R. Larson; B. Liu;R. Liu

    Comments

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    Abbreviated Journal Title

    J. Funct. Anal.

    Keywords

    Dilation; System of imprimitivity; Banach space; Projective isometric; representation; Operator-valued measure; Frame; FRAME REPRESENTATIONS; Mathematics

    Abstract

    Motivated by a general dilation theory for operator-valued measures, framings and bounded linear maps on operator algebras, we consider the dilation theory of the above objects with special structures. We show that every operator-valued system of imprimitivity has a dilation to a probability spectral system of imprimitivity acting on a Banach space. This completely generalizes a well-known result which states that every frame representation of a countable group on a Hilbert space is unitarily equivalent to a subrepresentation of the left regular representation of the group. We also prove that isometric group representation induced framings on a Banach space can be dilated to unconditional bases with the same structure for a larger Banach space. This extends several known results on the dilations of frames induced by unitary group representations on Hilbert spaces. (C) 2014 Elsevier Inc. All rights reserved.

    Journal Title

    Journal of Functional Analysis

    Volume

    266

    Issue/Number

    12

    Publication Date

    1-1-2014

    Document Type

    Article

    Language

    English

    First Page

    6914

    Last Page

    6937

    WOS Identifier

    WOS:000335935500007

    ISSN

    0022-1236

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