A II$_1$ factor approach to the Kadison-Singer problem

Abstract: We show that the Kadison-Singer problem, asking whether the pure states of the diagonal subalgebra $\ell^\infty\Bbb N\subset \Cal B(\ell^2\Bbb N)$ have unique state extensions to $\Cal B(\ell^2\Bbb N)$, is equivalent to a similar statement in II$_1$ factor framework, concerning the ultrapower inclusion $D^\omega \subset R^\omega$, where $D$ is the Cartan subalgebra of the hyperfinite II$_1$ factor $R$, and $\omega$ is a free ultraflter. While we do not settle the problem in this latter form, we prove that if $A$ is any singular maximal abelian subalgebra of $R$, then the inclusion $A^\omega \subset R^\omega$ does satisfy the Kadison-Singer property.