{"paper":{"title":"Bilinear forms and the $\\Ext^2$-problem in Banach spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Jes\\'us M. F. Castillo, Ricardo Garc\\'ia","submitted_at":"2018-08-09T14:01:35Z","abstract_excerpt":"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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.03173","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}