#### Vol. 8, No. 4, 2015

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Paving over arbitrary MASAs in von Neumann algebras

### Sorin Popa and Stefaan Vaes

Vol. 8 (2015), No. 4, 1001–1023
##### 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 $\mathsc{ℬ}\left({\ell }^{2}ℕ\right)$, 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.

##### Keywords
Kadison–Singer problem, paving, von Neumann algebra, maximal abelian subalgebra
##### Mathematical Subject Classification 2010
Primary: 46L10
Secondary: 46A22, 46L30