Simpliciality of vector-valued function spaces

TL;DR

This work develops a framework for integral representations of vector-valued function spaces $H\subset C(K,E)$, introducing two dual notions of simpliciality: vector simpliciality (via $E^*$-valued vector measures for functionals) and weak simpliciality (via scalar measures for operators). It shows these notions are generally incomparable, though when $H$ contains constants vector simpliciality is stronger, and it provides multiple characterizations and stability results under renorming. The paper also extends Choquet boundary theory to the vector-valued setting, defines $H$-boundary measures, and proves two representation theorems that connect to four notions of affine structure through $A_c^w(H)$ and $A_c^v(H)$. Additionally, it develops dilation tools, Dirichlet-type extension results, and a Batty-inspired ordering of measures, yielding detailed criteria for boundary measures and maximality. Together these results illuminate the structure and representation theory of vector-valued function spaces, with implications for functional-analytic and Choquet-theoretic analyses of $H$-boundary phenomena.

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.

Theorems & Definitions (133)

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• Example 2.3
• Lemma 3.1
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• Lemma 3.2
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• Example 3.3
• Lemma 3.4
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