Regular normed bimodules

TL;DR

This paper characterizes regular Banach bimodules over $C^*$-algebras of compact operators and (F)-Banach bundles over hyper-Stonian spaces, focusing on isometric properties of natural maps to biduals.

Contribution

It provides new characterizations of regular bimodules and (F)-Banach bundles based on isometric properties of natural maps, linking operator space theory and bundle theory.

Findings

Characterization of regular Banach bimodules over $C^*$-algebras of compact operators.

Identification of (F)-Banach bundles among all (H)-Banach bundles over hyper-Stonian spaces.

Conditions under which natural maps from bimodules to their biduals are isometric.

Abstract

In this article, we will give a characterization of Banach bimodules over $C^*$-algebras of compact operators that arises from operator spaces as well as a characterization of (F)-Banach bundles amongst all (H)-Banach bundles over a hyper-Stonian space. These two characterizations are concerned with whether certain natural map from a Banach bimodule to its canonical bidual is isometric (we call such bimodule \emph{regular}).