On C*-algebras cut down by closed projections: Characterizing elements viathe extreme boundary

Let A be a C*-algebra. Let z be the maximal atomic projection and p a close d projection in A**. It is known that x 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 pure states of A. Under some additional conditions, we shall show th at if x is uniformly continuous on the set of pure states of A supported by p, or its weak* closure, then pxp has a continuous atomic part, i.e., zpxp = zpap for some a in A.