The structure of standardC∗-algebras and their representations

Abstract

!• Introduction* The Wedderburn structure theorem asserts that every finite dimensional complex semi-simple algebra is a di rect sum of simple algebras, each of which is an n x n matrix algebra. The structure of a general infinite dimensional complex semi-simple algebra is certainly too complicated to study at present. Instead, we focus our attention on a well-behaved subclass of infinite dimensional algebras: C*-algebras. A C*-algebra A is a complex Banach *-algebra such that ||α*α|| = ||α|| 2 for all a in A. By using the Gelfand-Naimark-Segal construction, a C*-algebra can also be defined as a norm-closed self-adjoint subalgebra of B{H), the algebra of all bounded linear operators on a Hubert space H. There are structure theorems available for three families of C*-algebras: (A) the Gelfand-Naimark theorem for commutative C*-algebras, (B) von Neumann's direct integral decomposition theorem for von Neumann algebras on a separable Hubert space, (C) Kaplansky's structure theorem for liminal C*-algebras with Hausdorff spectrum which is actually a noncommutative generalization of (A) or can be viewed as a continuous generalization of the Wedderburn theorem. The main purpose of this paper is to prove a structure theorem for certain C*-algebras, called standard C*-algebras, which generalizes (C) as well as providing a continuous version of (B). Firstly, we introduce the basic structure space of a C*-algebra: A primitive ideal I of a C*-algebra A is called minimal if it does not contain any other primitive ideal of A. The basic structure space Ab of a C*-algebra A is the set of all minimal primitive ideals in A. A C*-algebra A is called bounded if every primitive ideal of A contains a minimal primitive ideal of A. It follows from the results of Kaplansky [23] and Dixmier [12] that type I C*-algebras and separable C*-algebras are bounded. We present another sufficient