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

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.