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arXiv · 1411.4559

Dilations of frames, operator valued measures and bounded linear maps

Deguang Han·David R. Larson·Bei Liu·Rui Liu
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Abstract

We will give an outline of the main results in our recent AMS Memoir, and include some new results, exposition and open problems. In that memoir we developed a general dilation theory for operator valued measures acting on Banach spaces where operator-valued measures (or maps) are not necessarily completely bounded. The main results state that any operator-valued measure, not necessarily completely bounded, always has a dilation to a projection-valued measure acting on a Banach space, and every bounded linear map, again not necessarily completely bounded, on a Banach algebra has a bounded homomorphism dilation acting on a Banach space. Here the dilation space often needs to be a Banach space even if the underlying space is a Hilbert space, and the projections are idempotents that are not necessarily self-adjoint. These results lead to some new connections between frame theory and operator algebras, and some of them can be considered as part of the investigation about "noncommutative" frame theory.

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Deguang Han, David R. Larson, Bei Liu, Rui Liu. 2014-11-17. Dilations of frames, operator valued measures and bounded linear maps. https://arxiv.org/abs/1411.4559

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