The β topology for W^∗-algebras

Jackson Neal Henry and Donald Curtis Taylor

Vol. 60 (1975), No. 1, 123–139

DOI: 10.2140/pjm.1975.60.123

Abstract

Let A be a W^∗-algebra and A∗ its unique predual. A new locally convex topology β is developed for the study of the algebra A. It is shown that if A is a type I W^∗-algebra, that is either countably decomposable, commutative, or a factor, then β is the Mackey topology for the dual pair ⟨A,A∗⟩. Consequently, when A = L^∞(X,μ), where X is completely regular and μ is a compact regular Borel measure on X, A_β^∗ = L¹(X,μ) and β convergence on uniformly bounded sets is equivalent to convergence in measure.

Mathematical Subject Classification 2000

Primary: 46L10

Milestones

Received: 5 March 1974

Published: 1 September 1975