Simpliciality of vector-valued function spaces

TL;DR

This paper explores two different generalizations of integral representation theorems for vector-valued function spaces, introducing notions of vector and weak simpliciality, and characterizes their properties and differences.

Contribution

It introduces and compares two new notions of simpliciality for vector-valued function spaces, providing characterizations and analyzing their invariance and implications.

Findings

Weak simpliciality is unaffected by renorming the target space.

Vector simpliciality is strictly stronger and has multiple characterizations.

The paper provides a refined representation theorem using positive measures.

Abstract

We investigate integral representation of vector-valued function spaces, i.e., of subspaces $H\subset C(K,E)$, where $K$ is a compact space and $E$ is a (real or complex) Banach space. We point out that there are two possible ways of generalizing representation theorems known from the scalar case -- either one may represent (all) functionals from $H^*$ using $E^*$-valued vector measures on $K$ (as it is done in the literature) or one may represent (some) operators from $L(H,E)$ by scalar measures on $K$ using the Bochner integral. These two ways lead to two different notions of simpliciality which we call `vector simpliciality' and `weak simpliciality'. It turns out that these two notions are in general incomparable. Moreover, the weak simpliciality is not affected by renorming the target space $E$, while vector simpliciality may be affected. Further, if $H$ contains constants, vector simpliciality is strictly stronger and admits several characterizations (partially analogous to the characterizations known in the scalar case). We also study orderings of measures inspired by C.J.K.~Batty which may be (in special cases) used to characterize $H$-boundary measures. Finally, we give a finer version of representation theorem using positive measures on $K\times B_{E^*}$ and characterize uniqueness in this case.