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.

Final version, to appear in Comm Math Phys: "Added in the Proof" at end of the Introd., on the recent solution to the classic Kadison-Singer by Marcus-Spielman-Strivastava (arXiv:1306.3969); one more characterization of singular MASAs in Thm. 0.2 (resp. Thm. 5.2); more details + complements in proof of 4.1; two sub-sections in Sec. 5 removed (to appear elsewhere)

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