On the Tensor Product of W ∗ Algebras

Abstract

We develop the algebra underlying the of von Neumann in the language and spirit of Sakai's abstract W* algebras, and using the maximum spectrum of an abelian von Neumann algebra rather than a measure-theoretic surrogate. We are thus enabled to obtain the basic fact of the von Neumann as a special case of a weaker general decomposition theorem, valid without separability or type restrictions, and adapted to comparison with Wright's in the finite case. Introduction. The study of von Neumann algebras (weakly closed rings of operators on Hilbert space) has always been largely algebraic. Many of their properties follow from the spectral theorem, and can consequently be developed for the abstract AW* algebras of Kaplansky [6], [7]. However there are some more delicate consequences of weak closure which do not survive in a general A W* algebra. For questions of this kind we have the striking abstract characterization due to Sakai [9]: a C* algebra is* isomorphic to a von Neumann algebra if and only if it is a dual Banach space. This is taken as the definition of a W* algebra in [11], where much of the standard of von Neumann algebras is developed in a space-free manner, including the so-called reduction theory of von Neumann [16]. The central algebraic fact is found to be a representation of the W* tensor product Z A of an abelian W* algebra Z = L??(F, i) with another W* algebra A, as the algebra Z = LX (F, jt; A) of all essentially bounded weakly* measurable A-valued functions on F, with pointwise operations. This result is restricted to the case when A can be faithfully represented on a separable Hilbert space because of measure-theoretic difficulties. In this paper we prove an analogous theorem, without separability hypotheses, by working with the continuous Gelfand representation Z = C(2) of the abelian algebra, rather than a measure-theoretic one, the role of null sets being played by meager sets (i.e., sets of the first category) in 2, as follows: Theorem. Let Z = C(u) be an abelian W* algebra, and let A be any W* algebra, with predual A*, and let B = Z @ A be the W* tensor product. Let C* (2, A) be the Banach space of all w* continuous functions from 2 to A, with the supremum norm. Then there is a natural isometry