Independence properties in subalgebras of ultraproduct II$_1$ factors

Abstract: 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 diffuse abelian von Neumann subalgebra in $\text{\bf Q}$ which is {\it free independent} to $X$, relative to $\text{\bf Q}'\cap \text{\bf M}$. Some related independence properties for subalgebras in ultraproduct II$_1$ factors are also discussed.