Paving over arbitrary masas in von neumann algebras

Abstract

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 work of Marcus, Spielman and Srivastava to check this for all MASAs in B(ℓ 2 N), 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 Cε -2, which we show to be sharp.