an:05241749
Zbl 1147.46016
Rodr??guez, Jos??
Weak Baire measurability of the balls in a Banach space
EN
Stud. Math. 185, No. 2, 169-176 (2008).
00216666
2008
j
46B26 28A05 28B05 46G10
Banach space; weak\(^*\)-separability; Baire \(\sigma\)-algebra; scalar measurability; Pettis integral
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 \textit{K.\,Musia??} [Rend. Ist. Mat. Univ. Trieste 23, 177--262 (1991; Zbl 0798.46042).
Hans Weber (Udine)
Zbl 0798.46042