Dilations of frames, operator valued measures and bounded linear maps

TL;DR

This paper develops a dilation theory for operator-valued measures and bounded linear maps on Banach spaces, extending classical results to non-completely bounded cases and exploring connections with noncommutative frame theory.

Contribution

It introduces a general dilation framework for operator-valued measures and linear maps on Banach spaces, including non-completely bounded cases, with applications to frame and operator algebra theory.

Findings

Any operator-valued measure has a dilation to a projection-valued measure on a Banach space.

Bounded linear maps on Banach algebras have bounded homomorphism dilations.

New links between frame theory and operator algebras are established.

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.