Paving over arbitrary MASAs in von Neumann algebras

Abstract

We consider a paving property for a maximal abelian $\ast$-subalgebra (MASA) $A$ in a von Neumann algebra $M$, that we call so-paving, involving approximation in the so-topology, rather than in norm (as in classical Kadison–Singer paving). If $A$ is the range of a normal conditional expectation, then so-paving is equivalent to norm paving in the ultrapower inclusion $A^{\omega }\subset M^{\omega }$. We conjecture that any MASA in any von Neumann algebra satisfies so-paving. We use work of Marcus, Spielman and Srivastava to check this for all MASAs in $\mathcal{ℬ}\left({\ell }^2\mathbb{N}\right)$, all Cartan subalgebras in amenable von Neumann algebras and in group measure space II${}_1$ factors arising from profinite actions. By earlier work of Popa, the conjecture also holds true for singular MASAs in II${}_1$ factors, and we obtain here an improved paving size $C{\epsilon }^{-2}$, which we show to be sharp.