Talagrand's Kσδ problem

Abstract We show that the Banach spaces C ( K ) with K either an adequate Talagrand compact or a quasi adequate σ -Eberlein Talagrand compact are K σ δ 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 K σ δ . We also provide an example of a Talagrand compact K such that C ( K ) is not K σ δ in its second dual. This answers a problem posed by M. Talagrand.