Independence properties in subalgebras of ultraproduct II₁ factors

2013/08/19 by Sorin Popa, Popa, Sorin · 3 citations
Mathematics · #46L10 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.1308.3982

openalex publication_date 2013/08/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let M_n be a sequence of finite factors with dim(M_n)→ ∞ and denote \bf M=Π_ωM_n their ultraproduct over a free ultrafilter ω. We prove that if \bf Q⊂ \bf M is either an ultraproduct \bf Q=Π_ωQ_n of subalgebras Q_n⊂ M_n, with Q_n \not\prec_Mn Q_n'∩ M_n, ∀ n, or the centralizer \bf Q=B'∩ \bf M of a separable amenable *-subalgebra B⊂ \bf M, then for any separable subspace X⊂ \bf M\ominus (\bf Q'∩ \bf M), there exists a diffuse abelian von Neumann subalgebra in \bf Q which is \it free independent to X, relative to \bf Q'∩ \bf M. Some related independence properties for subalgebras in ultraproduct II₁ factors are also discussed.

Citations

• On Groups of Measure Preserving Transformations. I

Cited by

• 50 Open Problems: Ultraproduct II₁ factors
• Compressible subalgebras in II₁ factors
• On the genericity of irreducible subfactors

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