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

We show that the Kadison-Singer problem, asking whether the pure states of the diagonal subalgebra ${\mathrm{\ell }}^{\mathrm{\infty }}\mathbb{N}\subset \text{Cal}B({\mathrm{\ell }}^2\mathbb{N})$ have unique state extensions to $\text{Cal}B({\mathrm{\ell }}^2\mathbb{N})$, 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.