{"paper":{"title":"Banach spaces whose algebra of bounded operators has the integers as their $K_0$-group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.OA"],"primary_cat":"math.KT","authors_text":"Niels Jakob Laustsen, Piotr Koszmider, Tomasz Kania","submitted_at":"2013-03-11T18:13:51Z","abstract_excerpt":"Let $X$ and $Y$ be Banach spaces such that the ideal of operators which factor through $Y$ has codimension one in the Banach algebra $\\mathscr{B}(X)$ of all bounded operators on $X$, and suppose that $Y$ contains a complemented subspace which is isomorphic to $Y\\oplus Y$ and that $X$ is isomorphic to $X\\oplus Z$ for every complemented subspace $Z$ of $Y$. Then the $K_0$-group of $\\mathscr{B}(X)$ is isomorphic to the additive group $\\mathbb{Z}$ of integers.\n  A number of Banach spaces which satisfy the above conditions are identified. Notably, it follows that $K_0(\\mathscr{B}(C([0,\\omega_1])))\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.2606","kind":"arxiv","version":2},"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"}