Weak Baire measurability of the balls in a Banach space
Let X be a Banach space. The property (?) “the unit ball of X belongs to Baire(X, weak)” holds whenever the unit ball of X∗ is weak∗separable; on the other hand, it is also known that the validity of (?) ensures that X∗ is weak∗-separable. In this paper we use suitable renormings of `∞(N) and the Johnson-Lindenstrauss spaces to show that (?) 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.