Remark on the Daugavet property for complex Banach spaces

In this article, we study the Daugavet property and the diametral diameter two properties (DD2Ps) in complex Banach spaces. The characterizations for both Daugavet and Δ-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 Δ-points from the fact that such spaces are locally uniformly nonsquare. We also study the convex diametral local diameter two property 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 A or the range space X X has the polynomial Daugavet property. Consequently, we obtain that the polynomial Daugavet property, Daugavet property, DD2Ps, and property (D) are equivalent for infinite-dimensional uniform algebras.