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A II$_1$ factor approach to the Kadison-Singer problem (1303.1424v3)

Published 6 Mar 2013 in math.OA

Abstract: We show that the Kadison-Singer problem, asking whether the pure states of the diagonal subalgebra $\ell\infty\Bbb N\subset \Cal B(\ell2\Bbb N)$ have unique state extensions to $\Cal B(\ell2\Bbb N)$, 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.

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