Bilinear forms and the $\Ext^2$-problem in Banach spaces

Abstract

Let $X$ be a Banach space and let $\kappa (X)$ denote the kernel of a quotient map ${\ell }_1(\Gamma )\to X$. We show that $\Ext^2(X,X^*)=0$ if and only if bilinear forms on $\kappa (X)$ extend to ${\ell }_1(\Gamma )$. From that we obtain i) If $\kappa (X)$ is a ${\mathcal{L}}_1$-space then $\Ext^2(X,X^*)=0$; ii) If $X$ is separable, $\kappa (X)$ is not an ${\mathcal{L}}_1$ space and $\Ext^2(X,X^*)=0$ then $\kappa (X)$ has an unconditional basis. This provides new insight into a question of Palamodov in the category of Banach spaces.