## Analysis & PDE

• Anal. PDE
• Volume 8, Number 4 (2015), 1001-1023.

### 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 $ℬ(ℓ2ℕ)$, 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.

#### Article information

Source
Anal. PDE, Volume 8, Number 4 (2015), 1001-1023.

Dates
Revised: 18 February 2015
Accepted: 25 March 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.apde/1510843118

Digital Object Identifier
doi:10.2140/apde.2015.8.1001

Mathematical Reviews number (MathSciNet)
MR3366008

Zentralblatt MATH identifier
1329.46056

#### Citation

Popa, Sorin; Vaes, Stefaan. Paving over arbitrary MASAs in von Neumann algebras. Anal. PDE 8 (2015), no. 4, 1001--1023. doi:10.2140/apde.2015.8.1001. https://projecteuclid.org/euclid.apde/1510843118

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