Analysis & PDE

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

Paving over arbitrary MASAs in von Neumann algebras

Sorin Popa and Stefaan Vaes

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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 work of Marcus, Spielman and Srivastava to check this for all MASAs in (2), all Cartan subalgebras in amenable von Neumann algebras and in group measure space II1 factors arising from profinite actions. By earlier work of Popa, the conjecture also holds true for singular MASAs in II1 factors, and we obtain here an improved paving size Cε2, which we show to be sharp.

Article information

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

Received: 12 January 2015
Revised: 18 February 2015
Accepted: 25 March 2015
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L10: General theory of von Neumann algebras
Secondary: 46A22: Theorems of Hahn-Banach type; extension and lifting of functionals and operators [See also 46M10] 46L30: States

Kadison–Singer problem paving von Neumann algebra maximal abelian subalgebra


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.

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