Paving over arbitrary MASAs in von Neumann algebras

I will present some recent work with Stefaan Vaes, in which we consider a paving property for a MASA $A$
in a von Neumann algebra $M$, that we call \emph{\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 recent work of Marcus-Spielman-Srivastava 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 work of mine from 2013, the conjecture also holds true for singular MASAs in II$_1$ factors, and we obtain an improved paving size
$C\varepsilon^{-2}$, which we show to be sharp.

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