Independence properties in subalgebras of ultraproduct II$_1$ factors
By Sorin Popa. 2013
50 citations recorded by the source corpus.
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.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/3ecdf3cf5f1e9f966d858062238fc0942d54a105
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- Fuglede–Kadison Determinants and Sofic Entropy (2014, background): “c groups contains all amenable groups, all residually finite groups, all linear groups and is closed under direct unions and free products with amalgamation over amenable subgroups (see [18],[16],[39],[41]). Residually sofic groups are also sofic. Thus sofic groups are a significantly larger class of gro”