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⊗ˆπY turns out to be a Daugavet point. We prove first that, under some hypothesis, the assumption of X⊗ˆπY having the Daugavet property implies the existence of a great amount of isometries from Y into X*. Having this in mind, we provide methods for constructing non-trivial Daugavet points in X⊗ˆπY⁠. We show that C(K)-spaces are examples of Banach spaces such that the set of the Daugavet points in C(K)⊗ˆπY is weakly dense when Y is a subspace of C(K)*. Finally, we present some natural results on when an elementary tensor x⊗y is a Daugavet point.