{"paper":{"title":"Harmonic cocycles, von Neumann algebras, and irreducible affine isometric actions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.GR","authors_text":"Bachir Bekka","submitted_at":"2016-12-28T18:27:23Z","abstract_excerpt":"Let $G$ be a compactly generated locally compact group and $(\\pi, \\mathcal H)$ a unitary representation of $G.$ The $1$-cocycles with coefficients in $\\pi$ which are harmonic (with respect to a suitable probability measure on $G$) represent classes in the first reduced cohomology $\\bar{H}^1(G,\\pi).$ We show that harmonic $1$-cocycles are characterized inside their reduced cohomology class by the fact that they span a minimal closed subspace of $\\mathcal H.$ In particular, the affine isometric action given by a harmonic cocycle $b$ is irreducible (in the sense that $\\mathcal H$ contains no non-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.08944","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"}