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Weak Baire measurability of the balls in a Banach space. (English) Zbl 1147.46016
Author’s abstract: Let $$X$$ be a Banach space. The property $$(\star)$$ “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 $$(\star)$$ 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. Musiał [Rend. Ist. Mat. Univ. Trieste 23, 177–262 (1991; Zbl 0798.46042).
Reviewer: Hans Weber (Udine)

MSC:
 46B26 Nonseparable Banach spaces 28A05 Classes of sets (Borel fields, $$\sigma$$-rings, etc.), measurable sets, Suslin sets, analytic sets 28B05 Vector-valued set functions, measures and integrals 46G10 Vector-valued measures and integration
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