Independence properties in subalgebras of ultraproduct II$_1$ factors

TL;DR

This paper investigates independence properties within subalgebras of ultraproduct II$_1$ factors, establishing conditions under which diffuse abelian subalgebras can be freely independent of certain subspaces.

Contribution

It introduces new independence results for subalgebras in ultraproduct II$_1$ factors, especially regarding free independence relative to commutants.

Findings

Existence of freely independent diffuse abelian subalgebras in ultraproducts.

Conditions under which subalgebras are free independent of given subspaces.

Discussion of related independence properties in 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.