Stability of diametral diameter two properties

We prove that the diametral diameter two properties are inherited by F -ideals (e.g., M -ideals). On the other hand, these properties are lifted from an M -ideal to the superspace under strong geometric assumptions. We also show that all of the diametral diameter two properties are stable under the formation of corresponding Köthe–Bochner spaces (e.g., L_p -Bochner spaces). Finally, we investigate when the projective tensor product of two Banach spaces has some diametral diameter two property.