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

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.