Paving over arbitrary MASAs in von Neumann algebras

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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 Aω⊂ Mω. We conjecture that any MASA in any von Neumann algebra satisfies so-paving. We use [MSS13] to check this for all MASAs in B(ℓ² N), all Cartan subalgebras in amenable von Neumann algebras and in group measure space II₁ factors arising from profinite actions. By [P13], the conjecture also holds true for singular MASAs in II₁ factors, and we obtain here an improved paving size Cε⁻², which we show to be sharp.