Remark on the Daugavet property for complex Banach spaces (Q6590327)

Let \(X\) be a Banach space and \(B_X\) be its unit ball. A slice of \(B_X\) is a non-empty set formed by the intersection of \(B_X\) with an open half-space. There are a number of Banach space properties that in the real case can be defined in terms of slices, and at the same time they have an alternative definition in terms of bounded linear operators. When one passes to complex spaces, the definition in terms of slices remains unchanged. The reformulation in terms of operators works if one speaks about real-linear operators, but for each of these properties there is the question of whether one can use complex-linear operators instead.\N\NThe article under review addresses several properties of such a kind in complex Banach spaces. A part of the paper is devoted to Delta-points and Daugavet points in complex Banach spaces, where the relationship between real and complex rank-one operators and rank-one projections comes in play. Another part deals with alternative convexity or smoothness, nonsquareness, and the anti-Daugavet property.\N\NThe last section of the paper deals with various properties like the Daugavet property, the convex diametral local diameter two property, the polynomial Daugavetian index, and the polynomial Daugavet property in the space \(A(K, X)\), which is an \(X\)-valued extension of a uniform algebra \(A\) on a compact \(K\). The central result of the section says that \(A(K, X)\) has the polynomial Daugavet property if and only if either the base algebra \(A\) or the range space \(X\) has the polynomial Daugavet property.