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 $$X=\mathcal {F}(\Omega )$$ X = F ( Ω ) is a Banach space of scalar-valued functions on a non-empty set $$\Omega $$ Ω and describe those spaces which admit a special isometric Banach predual, namely a strong isometric Banach linearisation , i.e. there are a Banach space Y , a map $$\delta :\Omega \rightarrow Y$$ δ : Ω → Y and an isometric isomorphism $$T:\mathcal {F}(\Omega )\rightarrow Y^{*}$$ T : F ( Ω ) → Y ∗ such that $$T(f)\circ \delta = f$$ T ( f ) ∘ δ = f for all $$f\in \mathcal {F}(\Omega )$$ f ∈ F ( Ω ) . Finally, we give necessary and sufficient conditions for Banach spaces $$\mathcal {F}(\Omega )$$ F ( Ω ) with a strong isometric Banach linearisation to have a (strongly) unique isometric Banach predual.