LEFT QUOTIENTS OF C∗-ALGEBRAS, II: ATOMIC PARTS OF LEFT QUOTIENTS

Abstract. Let A be a C∗-algebra. Let z be the maximal atomic projection in A∗∗. By a theorem of Brown, an element 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. Let L be a closed left ideal of A. Under some additional conditions, we shall show that if x is uniformly continuous on the set of pure states of A killing L, or its weak * closure, then x has a continuous atomic part modulo L∗ ∗ in an appropriate sense