ON C¤-ALGEBRAS CUT DOWN BY CLOSED PROJECTIONS: CHARACTERIZING ELEMENTS VIA THE EXTREME BOUNDARY

Abstract

Let A be a C¤ -algebra. Let z be the maximal atomic projection and p a closed 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 that 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.