Talagrand's Kσδ problem

We show that the Banach spaces C(K) with K either an adequate Talagrand compact or a quasi adequate σ-Eberlein Talagrand compact are Kσδ subsets of their second dual endowed with the weak∗ topology. As consequence we obtain that weakly K-analytic Banach spaces with an unconditional basis are Kσδ. We also provide an example of a Talagrand compact K such that C(K) is not Kσδ in its second dual. This answers a problem posed by M. Talagrand.