Finite sums of projections in von Neumann algebras

Halpern, Herbert; Kaftal, Victor; Ng, Ping; Zhang, Shuang

doi:10.1090/S0002-9947-2013-05683-8

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 $\tau (a_+)> \tau (a_-)$.

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