Positive combinations of projections in von Neumann algebras and purely   infinite simple C*-algebras
  We give an overview of the question: which positive elements in an operator algebra can be written as a linear combination of projections with positive coefficients. A special case of independent interest is the question of which positive elements can be written as a sum of finitely many projections. We focus on von Neumann algebras, on purely infinite simple C*-algebras, and on their associated multiplier algebras. 
V. Kaftal, P. W. Ng, S. Zhang
math.OA