Diffuse Sequences and Perfect C ∗ -Algebras

Abstract

The concept of a diffuse sequence in a C*-algebra is introduced and exploited to complete the classification of separable, perfect C*-algebras. A C*-algebra is separable and perfect exactly when the closure of the pure state space consists entirely of atomic states. 0. Introduction. In [3] the first author and F. Shultz introduced the notion of a perfect C*-algebra (see ?1 for the definition) and characterized separable, type I, perfect C*-algebras. In Theorem 3.11 we complete this classification by showing that no separable, nontype I C*-algebra is perfect. Our approach requires the introduction of the concept of a diffuse sequence {bn} of operators in a C*-algebra (see Definition 2.1). Such a sequence is considered to be trivial if lim IIbn = 0. In ?2 we develop the basic facts about diffuse sequences. As applications we show that the existence of nontrivial diffuse sequences is largely a phenomenon of separable C*-algebras with very non-Hausdorff spectrum. In particular, we show that neither separably represented von Neumann algebras nor corona algebras (M(A)/A, A aunital) can have nontrivial diffuse sequences. In ?3 we are aiming for the characterization of separable, perfect C*-algebras in Theorem 3.11, but our route takes us deeply into the Fermion algebra. A somewhat generalized theorem of Glimm [9, 6.7.3] allows us to do most of our specific construction in the Fermion algebra because we can use our lifting result, Proposition 2.11, to lift the constructed sequences to an arbitrary, separable, nontype I C*-algebra. 1. Notation and preliminaries. Generally we follow the notation of [9]. The letters A and B will always denote C*-algebras with elements a, b, c, d, e, p, q, r, s, u, v, w, x, y. The letters f, g, h will denote generic elements of A*, the dual space of A (with j, , and o used for some special elements). We shall frequently consider A as canonically embedded in its double dual A**, identified with the weak closure of A in its universal representation (see [9, p. 60]). For any elements a, b, c C A and f c A* define (afb) c A* by (afb)(c) = f(acb). Let S(A) denote the state space of A, Q(A) the quasi-state space of A, and P(A) the pure state space of A. Convergence in A* will default to weak* convergence, while the default convergence in A** is strong*. The letter z will be reserved for the central projection in A** covering the reduced atomic representation of A (see [9, p. 103]). Any g c Q(A) with g(z) = g(l) is called atomic while any f c Q(A) with f(z) = 0 is called diffuse, and, by [1, Lemma 1.3], Ilf gll = Ilf + gl. Received by the editors August 28, 1985 and, in revised form, January 10, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 46Lxx.