The super Alternative Daugavet property, unconditional bases and SCD geometry

Abstract

We answer negatively the 1-unconditional part of Question 6.4, by Langemets, L\~oo, Mart\'in, Perreau and Rueda Zoca, and, more generally, prove that no infinite-dimensional Banach space with a $K$-unconditional basis, for $1\leq K<3/2$, can satisfy the super Alternative Daugavet property. We also address two recent questions posed by L\~oo and Perreau concerning weak topological structures and slicely countably determined phenomena in Banach spaces with unconditional bases. More precisely, we prove that the weak unit ball of every Banach space with a 1-unconditional basis admits a countable $\pi$-base, thereby giving a positive answer to Question 5.1. We further show that every bounded convex subset of a Banach space with a Schauder basis that is shrinking or boundedly complete admits a countable $\pi$-base for its relative weak topology. We complement this result with two permanence principles: one for shrinking Schauder decompositions whose finite partial sums have countable weak $\pi$-bases, and another for unconditional sums over boundedly complete Banach sequence spaces. Finally, we prove that, for every $k>1$, there exists a Banach space with a $k$-unconditional basis whose unit ball is not slicely countably determined, thereby giving a positive answer to Question 5.4.