{"paper":{"title":"Unitarizability, Maurey--Nikishin factorization, and Polish groups of finite type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.GR"],"primary_cat":"math.OA","authors_text":"Andreas Thom, Asger T\\\"ornquist, Hiroshi Ando, Yasumichi Matsuzawa","submitted_at":"2016-05-23T07:02:43Z","abstract_excerpt":"Let $\\Gamma$ be a countable discrete group, and let $\\pi\\colon \\Gamma\\to {\\rm{GL}}(H)$ be a representation of $\\Gamma$ by invertible operators on a separable Hilbert space $H$. We show that the semidirect product group $G=H\\rtimes_{\\pi}\\Gamma$ is SIN ($G$ admits a two-sided invariant metric compatible with its topology) and unitarily representable ($G$ embeds into the unitary group $\\mathcal{U}(\\ell^2(\\mathbb N))$), if and only if $\\pi$ is uniformly bounded, and that $\\pi$ is unitarizable if and only if $G$ is of finite type: that is, $G$ embeds into the unitary group of a II$_1$-factor. Conse"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.06909","kind":"arxiv","version":3},"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"}