Independence properties in subalgebras of ultraproduct II${}_1$ factors

Abstract

Let $M_n$ be a sequence of finite factors with $\dim (M_n)\to \infty$ and denote $\text{M}={\Pi }_{\omega }{M}_n$ their ultraproduct over a free ultrafilter $\omega$. We prove that if $\text{Q}\subset \text{M}$ is either an ultraproduct $\text{Q}={\Pi }_{\omega }{Q}_n$ of subalgebras $Q_n\subset M_n$, with $Q_n/{\prec }_{M_n}{Q}_n^{\prime }\cap M_n$, $\forall n$, or the centralizer $\text{Q}=B^{\prime }\cap \text{M}$ of a separable amenable *-subalgebra $B\subset \text{M}$, then for any separable subspace $X\subset \text{M}\ominus ({\text{Q}}^{\prime }\cap \text{M})$, there exists a diffuse abelian von Neumann subalgebra in $\text{Q}$ which is {\it free independent} to $X$, relative to ${\text{Q}}^{\prime }\cap \text{M}$. Some related independence properties for subalgebras in ultraproduct II${}_1$ factors are also discussed.

Some minor corrections and additions; final version (33 pages), with corrections and 2.2.3 added; to appear in JFA

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• Free Entropy Dimension in Amalgamated Free Products
• About the Connes Embedding Conjecture---Algebraic approaches---