Vol. 8, No. 4, 2015

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Paving over arbitrary MASAs in von Neumann algebras

Sorin Popa and Stefaan Vaes

Vol. 8 (2015), No. 4, 1001–1023
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 (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.

Keywords
Kadison–Singer problem, paving, von Neumann algebra, maximal abelian subalgebra
Mathematical Subject Classification 2010
Primary: 46L10
Secondary: 46A22, 46L30
Milestones
Received: 12 January 2015
Revised: 18 February 2015
Accepted: 25 March 2015
Published: 21 June 2015
Authors
Sorin Popa
Department of Mathematics
University of California Los Angeles
Los Angeles, CA 90095-1555
United States
Stefaan Vaes
Department of Mathematics
KU Leuven
Celestijnenlaan 200B
3001 Leuven
Belgium