Remarks on the Daugavet Property for Complex Banach Spaces

TL;DR

This paper explores the Daugavet property and related diameter two properties in complex Banach spaces, providing new characterizations, relationships with convexity and smoothness, and conditions for polynomial Daugavet property in function spaces.

Contribution

It offers new characterizations of the Daugavet and Δ-points in complex Banach spaces and establishes conditions for the polynomial Daugavet property in vector-valued function spaces.

Findings

Strongly locally uniformly alternatively convex or smooth spaces lack Δ-points.

The polynomial Daugavet property in $A(K, X)$ depends on either $A$ or $X$ having it.

Various diameter two properties are equivalent in infinite-dimensional uniform algebras.

Abstract

In this article, we study the Daugavet property and the diametral diameter two properties in complex Banach spaces. The characterizations for both Daugavet and $\Delta$-points are revisited in the context of complex Banach spaces. We also provide relationships between some variants of alternative convexity and smoothness, nonsquareness, and the Daugavet property. As a consequence, every strongly locally uniformly alternatively convex or smooth (sluacs) Banach space does not contain $\Delta$-points from the fact that such spaces are locally uniformly nonsquare. We also study the convex diametral local diameter two property (convex-DLD2P) and the polynomial Daugavet property in the vector-valued function space $A(K, X)$. From an explicit computation of the polynomial Daugavetian index of $A(K, X)$, we show that the space $A(K, X)$ has the polynomial Daugavet property if and only if either the base algebra $A$ or the range space $X$ has the polynomial Daugavet property. Consequently, we obtain that the polynomial Daugavet property, the Daugavet property, the diameteral diameter two properties, and the property ($\mathcal{D}$) are equivalent for infinite-dimensional uniform algebras.