Bilinear forms and the Ext2-problem in Banach spaces

Abstract

Let X be a Banach space and let κ(X) denote the kernel of a quotient map ℓ1(Γ)→X. We show that Ext2(X,X⁎)=0 if and only if bilinear forms on κ(X) extend to ℓ1(Γ). From that we obtain i) If κ(X) is a L1-space then Ext2(X,X⁎)=0; ii) If X is separable, κ(X) is not an L1 space and Ext2(X,X⁎)=0 then κ(X) has an unconditional basis. This provides new insight into a question of Palamodov in the category of Banach spaces.