Unique Hahn-Banach extensions in Bochner $L^p$ spaces

Abstract

Let $Y$ be a closed subspace of a real Banach space $X$. Let $(Ω,Σ,μ)$ be a complete finite measure space and let $1<p<\infty$. We prove that $Y$ has property-$(U)$, property-$(SU)$, property-$(wU)$ in $X$ if and only if $L^p(μ,Y)$ has the corresponding property in $L^p(μ,X)$. No Radon-Nikodým assumption on $X^*$ is required. The proofs use $w^*$-measurable representations of dual functionals and a simultaneous scalar-lifting construction. A norm-minimal representation for an element in $L^p(μ,X)^*$ is found suitable in this case. We also study the point of continuity of the identity map under different topologies on the unit sphere of $L^p(μ,X)^*$. For Bochner-integrable functionals, the pointwise $w^*-w$-continuity yields the uniqueness of the extension to $L^p(μ,X^{**})$. Examples show why arbitrary $w^*$-measurable representatives cannot replace the norm-minimal representatives in this context.