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

30 pages, 1 figure