{"paper":{"title":"Flat bundles, von Neumann algebras and $K$-theory with $\\R/\\Z$-coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.KT"],"primary_cat":"math.OA","authors_text":"Georges Skandalis (IMJ), Paolo Antonini (IMJ), Sara Azzali (IMJ)","submitted_at":"2013-08-01T14:19:17Z","abstract_excerpt":"Let $M$ be a closed manifold and $\\alpha : \\pi_1(M)\\to U_n$ a representation. We give a purely $K$-theoretic description of the associated element $[\\alpha]$ in the $K$-theory of $M$ with $\\R/\\Z$-coefficients. To that end, it is convenient to describe the $\\R/\\Z$-$K$-theory as a relative $K$-theory with respect to the inclusion of $\\C$ in a finite von Neumann algebra $B$. We use the following fact: there is, associated with $\\alpha$, a finite von Neumann algebra $B$ together with a flat bundle $\\cE\\to M$ with fibers $B$, such that $E_\\a\\otimes \\cE$ is canonically isomorphic with $\\C^n\\otimes \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.0218","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"}