Attaining strong diameter two property for infinite cardinals

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 ω (1-ASD2Pω for short) if every convex series of slices of the unit ball intersects the unit sphere. We characterize C(K) spaces and L1(µ) spaces having the 1-ASD2Pω. We establish dual implications between the 1-ASD2Pω, ω-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. 2020 Mathematics Subject Classification. 46B04, 46B20.