On linearisation, existence and uniqueness of preduals: The isometric case

TL;DR

This paper investigates the conditions for the existence and uniqueness of isometric Banach preduals, especially for spaces of scalar-valued functions, and introduces the concept of strong isometric Banach linearisation.

Contribution

It provides necessary and sufficient conditions for isometric Banach preduals and characterizes spaces with strong isometric Banach linearisation.

Findings

Derived conditions for existence of isometric Banach preduals.

Characterized spaces admitting strong isometric Banach linearisation.

Established criteria for uniqueness of such preduals.

Abstract

We study the problem of existence and uniqueness of isometric Banach preduals of a Banach space. We derive necessary and sufficient conditions for the existence of an isometric Banach predual of a Banach space $X$. Then we focus on the case that $X=\mathcal{F}(\Omega)$ is a Banach space of scalar-valued functions on a non-empty set $\Omega$ and describe those spaces which admit a special isometric Banach predual, namely a \emph{strong isometric Banach linearisation}, i.e. there is a Banach space $Y$, a map $\delta\colon\Omega\to Y$ and an isometric isomorphism $T\colon\mathcal{F}(\Omega)\to Y^{\ast}$ such that $T(f)\circ \delta= f$ for all $f\in\mathcal{F}(\Omega)$. Finally, we give necessary and sufficient conditions for Banach spaces $\mathcal{F}(\Omega)$ with a strong isometric Banach linearisation to have a (strongly) unique isometric Banach predual.