Daugavet points in projective tensor products

Abstract

In this paper, we are interested in studying when an element $z$ in the projective tensor product $X{\hat{\otimes }}_{\pi }Y$ turns out to be a Daugavet point. We prove first that, under some hypothesis, the assumption of $X{\hat{\otimes }}_{\pi }Y$ having the Daugavet property implies the existence of a great amount of isometries from $Y$ into $X^{\ast }$. Having this in mind, we provide methods for constructing non-trivial Daugavet points in $X{\hat{\otimes }}_{\pi }Y$. We show that $C(K)$-spaces are examples of Banach spaces such that the set of the Daugavet points in $C(K){\hat{\otimes }}_{\pi }Y$ is weakly dense when $Y$ is a subspace of $C(K{)}^{\ast }$. Finally, we present some natural results on when an elementary tensor $x\otimes y$ is a Daugavet point.

16 pages