\(\sigma\)-fragmentability of multivalued maps and selection theorems.

*(English)*Zbl 0822.54018The authors answer the question as to when a weak-star upper semicontinuous map \(F\) with arbitrary non-empty values from a metric space \(T\) to the dual \(X^*\) of an Asplund Banach space \(X\) has a selector of the first Baire class to the norm. The existence of these selections has been successfully applied to several questions in the theory of Banach spaces and the paper contains also unified versions of the different resorts in the area. A version for set valued maps of the notion of \(\sigma\)-fragmentability considered in [the first author with I. Namioka and C. A. Rogers, Mathematika 39, 161-188 (1992; Zbl 0761.46008); ibid. 197-215 (1992; Zbl 0761.46009)] plays a central role here. A multivalued map \(F\) from a perfectly paracompact space \(T\) into the subsets of a metric space \((E, \rho)\) is \(\sigma\)-fragmented if and only if it can be uniformly approximated by first Baire class functions which are piecewise constant (theorem 5). If the map \(F\) has complete values and the \(\sigma\)-fragmentability is hereditary then \(F\) has a \(\sigma\)-discrete first Borel class selector (theorem 12). If \(T\) is a metric space, \(\rho\) is \(\tau\)-lower semicontinuous for a coarser topology \(\tau\) on \(E\) and \(F\) is \(\tau\)-upper semicontinuous with arbitrary non-void values the same \(\sigma\)-discrete first Borel class selector is obtained (theorem 13). This result contains previous ones by the first author and C. A. Rogers [Acta Math. 155, 41-79 (1985; Zbl 0588.54020)], R. W. Hansell [J. Funct. Anal. 75, 382-395 (1987; Zbl 0644.54014)], V. V. Srivatsa [Trans. Am. Math. Soc. 337, No. 2, 609-624 (1993; Zbl 0822.54017), see review above]. For Banach spaces they introduce new ideas and prove that a weak-star upper semicontinuous map \(F\) with arbitrary non-empty values from a metric space \(T\) to the dual \(X^*\) of an Asplund space \(X\) has a selector of the first Baire class provided that either \(X^*\) is weakly compactly generated or the unit ball \(B_{X^*}\) is angelic for the weak-star topology (i.e. if \(y\) is a point in the weak-star closure of a set \(A\subset B_{X^*}\) then there is a sequence in \(A\) weak-star convergent to \(y\)). Some counterexamples are also given to fix the sharpness of the results. Indeed, in every Asplund space \(X\) without the property \(C\) of Corson they construct weak-star upper semicontinuous set valued maps with non- empty, convex and weak-star countably compact values from \(X\) into \(X^*\) witout first Baire class selectors.

Reviewer: M.Valdivia (Valencia)