Stability Results of Diameter Two Properties

A Banach space has the diameter two property if every (nonempty) weakly open set of its unit ball has diameter two. We prove that this property is stable under finite sums, whenever an absolute norm is considered in the product space, improving some previous results. Recently T. A. Abrahamsen, V. Lima and O. Nygaard [J. Convex Analysis 20 (2013) 329–338] defined the so-called strong diameter two property, i.e. every convex combination of slices in the unit ball has diameter two. We show that the strong diameter two property is never stable under (non-trivial)

-sums. As a consequence, both diameter two properties are not equivalent. We also prove that any Banach space whose centralizer is infinite-dimensional has the strong diameter two property, solving an open problem. This result can be applied to infinite-dimensional

C^* C ∗

-algebras and

L_1 L 1

-preduals, for instance.