A II₁ Factor Approach to the Kadison–Singer Problem

Abstract

We show that the Kadison–Singer problem, asking whether the pure states of the diagonal subalgebra ${\ell^\infty\mathbb{N}\subset \mathcal{B}(\ell^2\mathbb{N})}$ have unique state extensions to ${\mathcal{B}(\ell^2\mathbb{N})}$, is equivalent to a similar statement in II₁ factor framework, concerning the ultrapower inclusion ${D^\omega \subset R^\omega}$, where D is the Cartan subalgebra of the hyperfinite II₁ factor R (i.e., a maximal abelian *-subalgebra of R whose normalizer generates R, e.g. ${D=L^\infty([0, 1]^{\mathbb{Z}}) \subset L^\infty([0,1]^{\mathbb{Z}} \rtimes \mathbb{Z} = R)}$, and ω is a free ultrafilter. Instead, we prove here that if A is any singular maximal abelian *-subalgebra of R (i.e., whose normalizer consists of the unitary group of A, e.g. ${A=L(\mathbb{Z})\subset L^\infty([0,1]^\mathbb{Z})\rtimes \mathbb{Z}=R}$), then the inclusion ${A^\omega \subset R^\omega}$ does satisfy the Kadison–Singer property.