Weakly open sets in the unit ball of the projective tensor product of Banach spaces

Abstract

A Banach space is said to have the diameter two property if every non-empty relatively weakly open subset of its unit ball has diameter two. We prove that the projective tensor product of two Banach spaces whose centralizer is infinite-dimensional has the diameter two property. The same statement also holds for X ⊗ ˆ π Y if the centralizer of X is infinite-dimensional and the unit sphere of Y ⁎ contains an element of numerical index one. We provide examples of classical Banach spaces satisfying the assumptions of the results. If K is any infinite compact Hausdorff topological space, then C ( K ) ⊗ ˆ π Y has the diameter two property for any nonzero Banach space Y. We also provide a result on the diameter two property for the injective tensor product.