Finite sums of projections in von Neumann algebras
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- by Herbert Halpern, Victor Kaftal, Ping Wong Ng and Shuang Zhang PDF
- Trans. Amer. Math. Soc. 365 (2013), 2409-2445 Request permission
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 $\|a\|_e$ 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 $\|a\|_e>1$, 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 $\tau (a_+)> \tau (a_-)$.References
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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
- MR Author ID: 96695
- Email: kaftalv@ucmail.uc.edu
- Ping Wong Ng
- Affiliation: Department of Mathematics, University of Louisiana, Lafayette, Louisiana 70504
- MR Author ID: 699995
- Email: png@louisiana.edu
- Shuang Zhang
- Affiliation: Department of Mathematics, University of Cincinnati, Cincinnati, Ohio 45221-0025
- Email: zhangs@ucmail.uc.edu
- Received by editor(s): July 27, 2010
- Received by editor(s) in revised form: August 11, 2011
- Published electronically: January 8, 2013
- © Copyright 2013 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 365 (2013), 2409-2445
- MSC (2010): Primary 47C15; Secondary 46L10
- DOI: https://doi.org/10.1090/S0002-9947-2013-05683-8
- MathSciNet review: 3020103