Finite sums of projections in von Neumann algebras

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. Then we 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 also for B(H). Thirdly, we prove that in a type II_1 factor a sufficient condition for a positive diagonalizable operators to be a finite sum of projections is that \tau(a_+)- \tau(a_-)>0.