Stability of diametral diameter two properties

TL;DR

This paper investigates the inheritance, lifting, and stability of diametral diameter two properties in Banach spaces, including their behavior under ideals, superspaces, K"othe-Bochner spaces, and tensor products.

Contribution

It establishes new results on how diametral diameter two properties are preserved or transferred across various Banach space constructions.

Findings

Diametral diameter two properties are inherited by F-ideals.

These properties are stable under K"othe-Bochner space formation.

Conditions are identified for the projective tensor product to have diametral diameter two properties.

Abstract

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 K\"othe-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.