A remark on the ultrapower algebra of the hyperfinite factor

TL;DR

This paper affirms a question posed by Sorin Popa regarding the structure of ultrapower algebras of hyperfinite II$_1$ factors, showing that certain inclusions imply equality of the factors.

Contribution

It provides a positive answer to Popa's question about ultrapower algebras, clarifying the structure of hyperfinite II$_1$ factors under specific conditions.

Findings

Confirmed Popa's conjecture on ultrapower algebras

Established conditions under which an inclusion of hyperfinite factors implies equality

Enhanced understanding of the structure of hyperfinite II$_1$ factors

Abstract

On page 43 in \cite{Po83} Sorin Popa asked whether the following property holds: \emph{If $\omega$ is a free ultrafilter on $\mathbb N$ and $\mathcal R_1\subseteq \mathcal R$ is an irreducible inclusion of hyperfinite II$_1$ factors such that $\mathcal R'\cap \mathcal R^\omega\subseteq \mathcal R^\omega_1$ does it follows that $\mathcal R_1=\mathcal R$?} In this short note we provide an affirmative answer to this question.