The Dual Space of an Operator Algebra

Introduction. The purpose of this paper is to study noncommutative C*algebras as Banach spaces. The Gelfand representation of an abelian C*-algebra as the algebra of all continuous complex-valued functions on its spectrum has made it possible to apply the techniques of measure theory and the topological properties of compact Hausdorff spaces to the study of such algebras. No such structure theory of general C*-algebras is available at present. Many theorems about the Banach space structure of abelian C*-algebras are stated in terms of topological or measure-theoretic properties of their spectra; although much work has been done of late in studying an analogous dual object for general C*-algebras, the generalization is far from exact. For this reason we shall confine our study primarily to W*-algebras in which the lattice of self-adjoint projections will be used as a substitute for the Borel sets of the spectrum of an abelian C*-algebra. Using a theorem of Takeda [15] we shall be able to extend some of our results to general C*-algebras. In [10] Sakai proved that any C*-algebra which is the dual of some Banach space has a representation as a H/*-aIgebra on some Hubert space. Dixmier [3] has proved the converse assertion, so it is possible to consider W*-algebras in a quite abstract fashion. It is this point of view which will predominate in this paper. Let F be a Banach space and suppose that the Banach space dual F* of F is a H/*-algebra, which will be denoted by M. In §1 we list some theorems and definitions about the topological properties of F and M as well as some related results. In §11 we give a number -of characterizations of the weakly relatively compact (abbreviated "wrc") subsets of F. These are applied to prove a conjecture of Sakai [13] that the Mackey topology of M agrees with the strong* topology on the unit sphere of M (see §1 for definitions). We conclude the section with an example which clearly shows the difference between the abelian and nonabelian cases and serves as a counterexample to other possible conjectures. In §111 we move to the dual M* of M where the situation becomes much more difficult. We are able to extend several of the characterizations of §11 to give conditions for weak relative compactness in M*. Also we give two formulations of the Vitali-Hahn-Saks Theorem for the noncommutative case and mention an open problem which remains in this area. Certain special algebras are the objects of study in §IV. We obtain a noncommutative version of Phillips' Lemma [9] for M/*-algebras with sufficiently many