Paving over arbitrary MASAs in von Neumann algebras

We consider a paving property for a maximal abelian *-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 [MSS13] to check this for all MASAs in $\mathcal B(\ell^2 \mathbb N)$, all Cartan subalgebras in amenable von Neumann algebras and in group measure space II$_1$ factors arising from profinite actions. By [P13], the conjecture also holds true for singular MASAs in II$_1$ factors, and we obtain here an improved paving size $C\varepsilon^{-2}$, which we show to be sharp.