Daugavet property in projective symmetric tensor products of Banach spaces

Abstract

We show that all the symmetric projective tensor products of a Banach space X have the Daugavet property provided X has the Daugavet property and either X is an L 1 -predual (i.e. X * is isometric to an L 1 -space) or X is a vector-valued L 1 -space. In the process of proving it, we get a number of results of independent interest. For instance, we characterise "localised" versions of the Daugavet property (i.e. Daugavet points and ∆-points introduced in [1]) for L 1 -preduals in terms of the extreme points of the topological dual, a result which allows to characterise a polyhedrality property of real L 1preduals in terms of the absence of ∆-points and also to provide new examples of L 1 -preduals having the convex diametral local diameter two property. These results are also applied to nicely embedded Banach spaces (in the sense of [40]) so, in particular, to function algebras. Next, we show that the Daugavet property and the polynomial Daugavet property are equivalent for L 1 -preduals and for spaces of Lipschitz functions. Finally, an improvement of recent results in [34] about the Daugavet property for projective tensor products is also got.