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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Towards the carpenter's theorem

Authors: Martín Argerami and Pedro Massey
Journal: Proc. Amer. Math. Soc. 137 (2009), 3679-3687
MSC (2000): Primary 46L99; Secondary 46L55
Published electronically: June 22, 2009
MathSciNet review: 2529874
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Abstract: Let $ \mathcal{M}$ be a II$ _1$ factor with trace $ \tau$, $ \mathcal{A}\subseteq \mathcal{M}$ a masa and $ E_{\mathcal{A}}$ the unique conditional expectation onto $ \mathcal{A}$. Under some technical assumptions on the inclusion $ \mathcal{A}\subseteq \mathcal{M}$, which hold true for any semiregular masa of a separable factor, we show that for elements $ a$ in certain dense families of the positive part of the unit ball of $ \mathcal{A}$, it is possible to find a projection $ p\in\mathcal{M}$ such that $ E_{\mathcal{A}}(p)=a$. This shows a new family of instances of a conjecture by Kadison, the so-called ``carpenter's theorem''.

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Additional Information

Martín Argerami
Affiliation: Department of Mathematics, University of Regina, Regina Saskatchewan, Canada

Pedro Massey
Affiliation: Departamento de Matemática, Universidad Nacional de La Plata and Instituto Argentino de Matemática-conicet, Argentina

PII: S 0002-9939(09)09999-7
Keywords: Diagonals of operators, Schur-Horn theorem, conditional expectations
Received by editor(s): July 17, 2007
Published electronically: June 22, 2009
Additional Notes: The first author was supported in part by the Natural Sciences and Engineering Research Council of Canada
The second author was supported in part by CONICET of Argentina, UNLP, and a PIMS Postdoctoral Fellowship
Communicated by: Marius Junge
Article copyright: © Copyright 2009 American Mathematical Society