Remarks on the Daugavet Property for Complex Banach Spaces

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.