Daugavet property in projective symmetric tensor products of Banach spaces

TL;DR

This paper demonstrates that symmetric projective tensor products of Banach spaces inherit the Daugavet property under certain conditions, and explores related geometric properties and equivalences in Banach space theory.

Contribution

It establishes the inheritance of the Daugavet property in symmetric projective tensor products for specific Banach spaces and characterizes localised Daugavet points in $L_1$-preduals.

Findings

Symmetric projective tensor products of Banach spaces with the Daugavet property also have it.

Characterization of Daugavet points and $ riangle$-points in $L_1$-preduals.

Equivalence of Daugavet and polynomial Daugavet properties in certain 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 $\Delta$-points recently introduced) 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 $L_1$-preduals in terms of the absence of $\Delta$-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 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 about the Daugavet property for projective tensor products is also got.