Dilations for Systems of Imprimitivity acting on Banach Spaces

TL;DR

This paper develops a dilation theory for operator-valued systems of imprimitivity on Banach spaces, generalizing known Hilbert space results and providing new dilation methods for frames and group representations.

Contribution

It extends dilation results from Hilbert spaces to Banach spaces, including positive operator systems and isometric group representations, broadening the scope of dilation theory.

Findings

Every operator-valued system of imprimitivity can be dilated to a probability spectral system on a Banach space.

Positive operator-valued systems of imprimitivity can be dilated to Hilbert spaces.

Isometric group representation induced framings can be dilated to unconditional bases on larger Banach spaces.

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.