Paving over arbitrary MASAs in von Neumann algebras
Presenter
March 12, 2015
Keywords:
- Banach algebras
- compact operators
- functional analysis
- operator algebras and ideals
- Kadison-Singer theorem
- Marcus-Spielman-Srivastava theorem
MSC:
- 46-xx
- 46Lxx
- 46L52
- 46L54
- 46L10
Abstract
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 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.