Almost squareness and strong diameter two property in tensor product spaces

Abstract

We study almost squareness and the strong diameter two property in the setting of projective (symmetric) tensor product of Banach spaces. We prove that almost squareness is stable by taking projective tensor products, providing non-trivial examples of ASQ projective tensor products of Banach spaces. Furthermore, we give sufficient conditions for a projective symmetric tensor product to have the strong diameter two property. This extend most of the previously known results and provide new examples of projective symmetric tensor product spaces with the strong diameter two property.