Banach spaces with the weak diametral diameter two property

Abstract

We introduce and systematically study the weak diametral diameter two property (weak-DD2P), a new geometric property that lies strictly between the diametral diameter two property and both the diameter two property and the convex diametral local diameter two property. A necessary condition for the weak-DD2P is obtained through the structure of the extreme points of the dual unit ball, which leads to characterisations for several classical classes of spaces, including $C(K)$ spaces, $L_1$-preduals, unital uniform algebras, and (vector-valued) function algebras. We establish stability results under standard constructions such as absolute sums, Köthe--Bochner spaces, and projective (symmetric) tensor products. Moreover, we provide complete descriptions of the weak-DD2P for vector-valued spaces of the form $L_1(μ,X)$, $L_\infty(μ,X)$, and $C(K,X)$. These results yield a wide range of new examples and show that the weak-DD2P exhibits a behaviour genuinely different from that of other diameter two properties.

Disclosure

“u” Excellence Unit IMAG (CEX2020-001105-M) funded by MICIU/AEI/10.13039/501100011033 and by Junta de Andalucı́a, and by Junta de Andalucı́a grant FQM-185. The authors acknowledge the use of artificial intelligence–based tools, including Microsoft Copilot 365, ChatGPT (OpenAI), and Gemini (Google), for language editing and stylistic improvements of the manuscript. The authors remain fully responsible for the content of this work. Reference”

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2026-08-24T00:34:05.733Z

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