Finite sums of projections in purely infinite simple C*-algebras with   torsion K_0
  Assume that A is a purely infinite simple C*-algebra whose K_0 is a torsion group, namely, contains no free element. Then a positive element a in A can be written as a finite sum of projections in A if and only if either a is a projection or the norm ||a||>1. 
V. Kaftal, P. N. Ng, S. Zhang
math.OA