On commutative V*-algebras

Abstract

We shall use results of Palmer (10, 11) and of Edwards and Ionescu Tulcea (6) to show that a commutative V*-algebra (with identity) of operators on a weakly complete Banach space is isomorphic to such an algebra on a Hilbert space, the isomorphism extending to the weak closures of the algebras. This result leads to an extension of Stone's theorem on unitary groups (a similar extension is proved by different methods in (2, p. 350) and of Nagy's theorems on semigroups of normal operators. The same technique yields an easy proof of Dunford's theorem on the existence of a σ-complete extension of a bounded Boolean algebra of projections on a weakly complete Banach space. We are indebted to H. R. Dowson for suggesting this topic and for help and guidance in pursuing it.