A weak Asplund space whose dual is not weak\(^*\) fragmentable (Q2750874)

A Banach space \(X\) is weak Asplund if any continuous convex function defined on a convex open subset \(A\) of \(X\) is Gâteaux differentiable at the points of a residual subset of \(A\). The authors construct, under some additional axioms of set theory (namely, their construction works under \(\omega_1=\omega_1^L\)), an example of a weak Asplund space \(X\) whose dual equipped with the weak* topology is not fragmentable. This solves a long-standing problem mentioned for example by \textit{M. J.Fabian} in his book [``Gâteaux differentiability of convex functions and topology. Weak Asplund spaces'' (1997; Zbl 0883.46011)]. NEWLINENEWLINENEWLINEIn fact, the dual \(X^*\) belongs to Stegall's class of spaces. The space \(X\) is of the form \(C(K)\), where \(K\) is the Stegall compact space which is not fragmentable constructed previously by the reviewer [Topology Appl. 96, No. 2, 121-133 (1999; Zbl 0991.54030)]. NEWLINENEWLINENEWLINEReviewer's remark: Using the results of this paper the reviewer recently showed [Proc. Am. Math. Soc. 130, No. 7, 2139-2143 (2002; review above)], under some other additional axioms of set theory, that there is a weak Asplund space whose dual does not even belong to Stegall's class of spaces.