Attaining strong diameter two property for infinite cardinals

TL;DR

This paper extends the strong diameter two property to infinite cardinals, characterizing certain Banach spaces and exploring the relationships and stability of these properties under various operations.

Contribution

It introduces the 1-ASD2P_ω property for infinite cardinals, characterizes spaces with this property, and studies its implications and stability in Banach space theory.

Findings

Characterization of C(K) and L_1(μ) spaces with 1-ASD2P_ω

Dual implications between 1-ASD2P_ω, ω-octahedral norms, and the (-1)-ball-covering property

Stability results under direct sums and tensor products

Abstract

We extend the (attaining of) strong diameter two property to infinite cardinals. In particular, a Banach space has the 1-norming attaining strong diameter two property with respect to $\omega$ (1-ASD2P$_\omega$ for short) if every convex series of slices of the unit ball intersects the unit sphere. We characterize $C(K)$ spaces and $L_1(\mu)$ spaces having the 1-ASD2P$_\omega$. We establish dual implications between the 1-ASD2P$_\omega$, $\omega$-octahedral norms and Banach spaces failing the $(-1)$-ball-covering property. The stability of these new properties under direct sums and tensor products is also investigated.