Baire classes of complex \(L_1\)-preduals. (English) Zbl 1363.46019
Summary: Let \(X\) be a complex \(L_1\)-predual, non-separable in general. We investigate extendability of complex-valued bounded homogeneous Baire-\(\alpha \) functions on the set \(\operatorname{ext}B_{X^*}\) of the extreme points of the dual unit ball \(B_{X^*}\) to the whole unit ball \(B_{X^*}\). As a corollary we show that, given \(\alpha \in [1,\omega_1)\), the intrinsic \(\alpha \)-th Baire class of \(X\) can be identified with the space of bounded homogeneous Baire-\(\alpha \) functions on the set \(\operatorname{ext} B_{X^*}\) when \(\operatorname{ext} B_{X^*}\) satisfies certain topological assumptions. The paper is intended to be a complex counterpart to the same authors’ paper: [“Baire classes of non-separable \(L_1\)-preduals”, Q. J. Math. 66, No. 1, 251–263 (2015; Zbl 1326.46015)]. As such it generalizes former work of J. Lindenstrauss and D. E. Wulbert [J. Funct. Anal. 4, 332–349 (1969; Zbl 0184.15102)], F. Jellett [Q. J. Math., Oxf. II. Ser. 36, 71–73 (1985; Zbl 0582.46010)].
MSC:
| 46B25 | Classical Banach spaces in the general theory |
| 26A21 | Classification of real functions; Baire classification of sets and functions |
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