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

 

Finite sums of projections in von Neumann algebras


Authors: Herbert Halpern, Victor Kaftal, Ping Wong Ng and Shuang Zhang
Journal: Trans. Amer. Math. Soc. 365 (2013), 2409-2445
MSC (2010): Primary 47C15; Secondary 46L10
Published electronically: January 8, 2013
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Abstract: We first prove that in a $ \sigma $-finite von Neumann factor $ M$, a positive element $ a$ with properly infinite range projection $ R_a$ is a linear combination of projections with positive coefficients if and only if the essential norm $ \Vert a\Vert _e$ with respect to the closed two-sided ideal $ J(M)$ generated by the finite projections of $ M$ does not vanish. We then show that if $ \Vert a\Vert _e>1$, then $ a$ is a finite sum of projections. Both these results are extended to general properly infinite von Neumann algebras in terms of central essential spectra. Secondly, we provide a necessary condition for a positive operator $ a$ to be a finite sum of projections in terms of the principal ideals generated by the excess part $ a_+:=(a-I)\chi _a(1,\infty )$ and the defect part $ a_-:= (I-a)\chi _a(0, 1)$ of $ a$; this result appears to be new for $ B(H)$ also. Thirdly, we prove that in a type II$ _1$ factor a sufficient condition for a positive diagonalizable operator to be a finite sum of projections is that $ \tau (a_+)> \tau (a_-)$.


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Additional Information

Herbert Halpern
Affiliation: Department of Mathematics, University of Cincinnati, Cincinnati, Ohio 45221-0025
Email: halperhp@ucmail.uc.edu

Victor Kaftal
Affiliation: Department of Mathematics, University of Cincinnati, Cincinnati, Ohio 45221-0025
Email: kaftalv@ucmail.uc.edu

Ping Wong Ng
Affiliation: Department of Mathematics, University of Louisiana, Lafayette, Louisiana 70504
Email: png@louisiana.edu

Shuang Zhang
Affiliation: Department of Mathematics, University of Cincinnati, Cincinnati, Ohio 45221-0025
Email: zhangs@ucmail.uc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05683-8
PII: S 0002-9947(2013)05683-8
Keywords: Finite sums of projections, positive combinations of projections, essential central spectrum
Received by editor(s): July 27, 2010
Received by editor(s) in revised form: August 11, 2011
Published electronically: January 8, 2013
Article copyright: © Copyright 2013 American Mathematical Society