  In this paper we present new structural information about the multiplier
algebra Mult (A) of a sigma-unital purely infinite simple C*-algebra A, by
characterizing the positive elements a in Mult(A) that are strict sums of
projections belonging to A. If a is not in A and is not a projection, then the
necessary and sufficient condition for a to be a strict sum of projections
belonging to A is that the norm ||a||>1 and that the essential norm ||a||_ess
>=1.
  Based on a generalization of the Perera-Rordam weak divisibility of separable
simple C*-algebras of real rank zero to all sigma-unital simple C*-algebras of
real rank zero, we show that every positive element of A with norm greater than
1 can be approximated by finite sums of projections. Based on block
tri-diagonal approximations, we decompose any positive element a in Mult(A)
with ||a||>1 and ||a||_ess >=1 into a strictly converging sum of positive
elements in A with norm greater than 1.

Operator Algebras;Functional Analysis