Left quotients of C*-algebras, II: Atomic parts of left quotients

Let A be a C*-algebra. Let z be the maximal atomic projection in A**. By a theorem of Brown, an element z in A** has a continuous atomic part, i.e. zx = za for some a in A, whenever x is uniformly continuous on the set of pur e states of A. Let L be a closed left ideal of A. Under some additional con ditions, we shall show that if lr is uniformly continuous on the set of pur e states of A killing L, or its weak* closure, then x has a continuous atom ic part module L** in an appropriate sense.