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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Commutator estimates in $ W^*$-factors


Authors: A. F. Ber and F. A. Sukochev
Journal: Trans. Amer. Math. Soc. 364 (2012), 5571-5587
MSC (2010): Primary 46L57, 46L51, 46L52
Published electronically: May 21, 2012
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Abstract: Let $ \mathcal {M}$ be a $ W^*$-factor and let $ S\left ( \mathcal {M} \right ) $ be the space of all measurable operators affiliated with $ \mathcal {M}$. It is shown that for any self-adjoint element $ a\in S(\mathcal {M})$ there exists a scalar $ \lambda _0\in \mathbb{R}$, such that for all $ \varepsilon > 0$, there exists a unitary element $ u_\varepsilon $ from $ \mathcal {M}$, satisfying $ \vert[a,u_\varepsilon ]\vert \geq (1-\varepsilon )\vert a-\lambda _0\mathbf {1}\vert$. A corollary of this result is that for any derivation $ \delta $ on $ \mathcal {M}$ with the range in an ideal $ I\subseteq \mathcal {M}$, the derivation $ \delta $ is inner, that is, $ \delta (\cdot )=\delta _a(\cdot )=[a,\cdot ]$, and $ a\in I$. Similar results are also obtained for inner derivations on $ S(\mathcal {M})$.


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Additional Information

A. F. Ber
Affiliation: Department of Mathematics, Tashkent State University, Uzbekistan
Email: ber@ucd.uz

F. A. Sukochev
Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia
Email: f.sukochev@unsw.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05568-1
PII: S 0002-9947(2012)05568-1
Keywords: Derivations in von Neumann algebras, measurable operators, ideals of compact operators
Received by editor(s): November 18, 2010
Received by editor(s) in revised form: February 15, 2011
Published electronically: May 21, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.