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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 \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 Δ \Delta -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 ) A\left(K,X) . From an explicit computation of the polynomial Daugavetian index of A ( K , X ) A\left(K,X) , we show that the space A ( K , X ) A\left(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 {\mathcal{D}} ) are equivalent for infinite-dimensional uniform algebras.
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 \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 Δ \Delta -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 ) A\left(K,X) . From an explicit computation of the polynomial Daugavetian index of A ( K , X ) A\left(K,X) , we show that the space A ( K , X ) A\left(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 {\mathcal{D}} ) are equivalent for infinite-dimensional uniform algebras.
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