Weak operator Daugavet property and weakly open sets in tensor product spaces

TL;DR

This paper advances understanding of the diameter two and Daugavet properties in tensor product spaces, establishing new conditions under which these properties hold, thus improving and extending previous results.

Contribution

It introduces new conditions involving the weak operator Daugavet property (WODP) that ensure diameter two and Daugavet properties in tensor products, advancing the theory.

Findings

If $X^*$ has the WODP, then $X\widehat{\otimes}_\varepsilon Y$ has the DD2P for any Banach space $Y$.

If $X$ has the WODP, then $X\widehat{\otimes}_\pi Y$ has the DD2P for any Banach space $Y$.

If $X^*$ and $Y^*$ have the WODP, then $X\widehat{\otimes}_\varepsilon Y$ has the Daugavet property.

Abstract

We obtain new progresses about the diameter two property and the Daugavet property in tensor product spaces. Namely, the main results of the paper are: -If $X^*$ has the WODP, then $X\widehat{\otimes}_\varepsilon Y$ has the DD2P for any Banach space $Y$. -If $X$ has the WODP, then $X\widehat{\otimes}_\pi Y$ has the DD2P for any Banach space $Y$. -If $X^*$ and $Y^*$ have the WODP then $X\widehat{\otimes}_\varepsilon Y$ has the Daugavet property. The above improve many results in the literature and establish progresses on some open questions.