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arXiv · 1411.4301

Dilations for Systems of Imprimitivity acting on Banach Spaces

Deguang Han·David R. Larson·Bei Liu·Rui Liu
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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-kown 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. The dilated space in general can not be taken as a Hilbert space. However, it can be taken as a Hilbert space for positive operator valued systems of imprimitivity. 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.

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Deguang Han, David R. Larson, Bei Liu, Rui Liu. 2014-11-16. Dilations for Systems of Imprimitivity acting on Banach Spaces. https://doi.org/10.1016/j.jfa.2014.02.040

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