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Paving over arbitrary
MASAs in von Neumann algebras
Sorin Popa and Stefaan Vaes |
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Vol. 8 (2015), No. 4, 1001–1023
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Abstract |
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We consider a paving property for a maximal abelian -subalgebra (MASA) in a von Neumann algebra , that we call so-paving, involving approximation in the so-topology, rather than in norm (as in classical Kadison–Singer paving). If is the range of a normal conditional expectation, then so-paving is equivalent to norm paving in the ultrapower inclusion . 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 , all Cartan subalgebras in amenable von Neumann algebras and in group measure space II factors arising from profinite actions. By earlier work of Popa, the conjecture also holds true for singular MASAs in II factors, and we obtain here an improved paving size , which we show to be sharp. |
Keywords
Kadison–Singer problem, paving, von Neumann algebra,
maximal abelian subalgebra
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Mathematical Subject Classification 2010
Primary: 46L10
Secondary: 46A22, 46L30
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Milestones
Received: 12 January 2015
Revised: 18 February 2015
Accepted: 25 March 2015
Published: 21 June 2015
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Authors |
| Sorin Popa | |
| Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555 United States |
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| Stefaan Vaes | |
| Department of Mathematics KU Leuven Celestijnenlaan 200B 3001 Leuven Belgium |
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