The dual spaces of 𝐶*-algebras

Abstract

Introduction. The idea of the structure space (or dual space) A of an associative algebra A was introduced by Jacobson in [8]. The space A consists of all kernels of irreducible representations of A, with the hull-kernel topology: An ideal I in A is in the closure of a subset B of A if I contains the intersection of the ideals in B. For unrestricted infinite-dimensional A, the dual space need not be Hausdorff or even T1; and in many situations it is not very useful. However, Gelfand and others have shown that for commutative Banach algebras the dual space is a powerful tool. For noncommutative Banach algebras, too, the study of the dual space has been found fruitful. Kaplansky [12] has analyzed the dual spaces of C*-algebras whose irreducible *-representations all consist of completely continuous operators. The importance of this study is emphasized by the fact that the group algebras of connected semi-simple Lie groups having faithful matrix representations all fall into this category (see [7]). This paper deals with some questions concerning the dual spaces of noncommutative C*-algebras, especially the group C*-algebras of certain groups. The contents of its three chapters are as follows: Chapter I centers around the equivalence theorem (Theorem 1.2) (2). This is a theorem specifically about C*-algebras. It states that, if 8 is a family of *-representations of a C*-algebra A, and T is a *-representation of A which vanishes for those elements for which all S in 8 vanish, then positive functionals associated with T are weakly* approximated by sums of positive functionals associated with S. In another form, it states a one-to-one correspondence between closed two-sided ideals of a C*-algebra and certain subsets of the positive cone of its conjugate space. In the latter form, the theorem was communicated to this author by R. Prosser, who also suggested the short proof of Theorem 1.1 given here. An interesting corollary of this theorem is the following: If G is a locally compact group, the hull-kernel topology of the dual space of its group C*-algebra is equivalent to the topology which Godement defined in [5] for the space e of irreducible unitary representations of G, using functions of positive type. Let us refer to this simply as the topology of L.