Weak Baire measurability of the balls in a Banach space
Volume 185 / 2008
Studia Mathematica 185 (2008), 169-176
MSC: 28A05, 28B05, 46B20, 46G10.
DOI: 10.4064/sm185-2-5
Abstract
Let $X$ be a Banach space. The property $(\star)$ “the unit ball of $X$ belongs to ${\rm Baire}(X,{\rm weak})$” holds whenever the unit ball of $X^{*}$ is weak$^{*}$-separable; on the other hand, it is also known that the validity of $(\star)$ ensures that $X^{*}$ is weak$^{*}$-separable. In this paper we use suitable renormings of $\ell^{\infty}(\mathbb{N})$ and the Johnson–Lindenstrauss spaces to show that $(\star)$ lies strictly between the weak$^{*}$-separability of $X^{*}$ and that of its unit ball. As an application, we provide a negative answer to a question raised by K. Musia/l.
