  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.
