On various diametral notions of points in the unit ball of some vector-valued function spaces

TL;DR

This paper investigates various types of diametral points in the unit balls of vector-valued function spaces, providing characterizations and stability results, especially highlighting equivalences in certain spaces.

Contribution

It offers new characterizations and stability results for diametral points in vector-valued function spaces, clarifying their relationships and properties.

Findings

Seven notions of diametral points are equivalent for $L_()$ and uniform algebras with infinite $K$.

Improved stability results under $_$ and $_$-sums.

$ abla$ points coincide with Daugavet points in complex Banach spaces.

Abstract

In this article, we study the ccs-Daugavet, ccs-$\Delta$, super-Daugavet, super-$\Delta$, Daugavet, $\Delta$, and $\nabla$ points in the unit balls of vector-valued function spaces $C_0(L, X)$, $A(K, X)$, $L_\infty(\mu, X)$, and $L_1(\mu, X)$. To partially or fully characterize these diametral points, we first provide improvements of several stability results under $\oplus_\infty$ and $\oplus_1$-sums shown in the literature. For complex Banach spaces, $\nabla$ points are identical to Daugavet points, and so the study of $\nabla$ points only makes sense when a Banach space is real. Consequently, we obtain that the seven notions of diametral points are equivalent for $L_\infty(\mu)$ and uniform algebra when $K$ is infinite.