{"paper":{"title":"Independence properties in subalgebras of ultraproduct II$_1$ factors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Sorin Popa","submitted_at":"2013-08-19T11:06:15Z","abstract_excerpt":"Let $M_n$ be a sequence of finite factors with $\\dim(M_n)\\rightarrow \\infty$ and denote $\\text{\\bf M}=\\Pi_\\omega M_n$ their ultraproduct over a free ultrafilter $\\omega$. We prove that if $\\text{\\bf Q}\\subset \\text{\\bf M}$ is either an ultraproduct $\\text{\\bf Q}=\\Pi_\\omega Q_n$ of subalgebras $Q_n\\subset M_n$, with $Q_n \\not\\prec_{M_n} Q_n'\\cap M_n$, $\\forall n$, or the centralizer $\\text{\\bf Q}=B'\\cap \\text{\\bf M}$ of a separable amenable *-subalgebra $B\\subset \\text{\\bf M}$, then for any separable subspace $X\\subset \\text{\\bf M}\\ominus (\\text{\\bf Q}'\\cap \\text{\\bf M})$, there exists a diffus"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.3982","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"}