Dilations for systems of imprimitivity acting on Banach spaces
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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. © 2014 Elsevier Inc.
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Han, D., Larson, D. R., Liu, B., & Liu, R.
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Han, Deguang||Larson, David R||Liu, Bei||Liu, Rui
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Banach Space
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Dilation
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Frame
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Operator-valued Measure
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Projective Isometric Representation
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System Of Imprimitivity
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