A Banach space with an unconditional basis which is not slicely countably determined

Abstract

In this note, we study the geometry of the unit ball of the Banach space generated by the adequate family of all subsets of branches of the infinite binary tree, and answer several open questions related to slicely countably determined Banach spaces. Our main result is that the binary tree space is an example of a Banach space with an unconditional basis which fails to be slicely countably determined. In particular, it provides an example of a non slicely countably determined separable Banach space which contains no isomorphic copy of a space with the Daugavet property. We also exhibit some other geometric features of this space: we prove that its unit ball is dentable, that it has numerical index~1, and that the points of continuity of its unit ball form a weakly dense set. Finally, we show that the binary tree space contains a non-convex subset which is slicely countably determined, but does not admit a countable $\pi$-base for its relative weak topology, and that there is a 2-equivalent renorming of this space whose unit ball fails to be slicely countably determined.