A remark on W*-tensor products of W*-Algebras

Abstract

Let E be a W*-algebra, T a hyperstonian compact space, C (T) the W*- algebra of continuous scalar valued functions on T, and F(T,E) the set of bounded maps x: T → E such that for every element a of the predual of E the function (Formula presented)is continuous. We define for every x ∈ F(T, E) an element x ∈ C (T)⊗ E such that the map (Formula presented) is a bijective isometry of ordered involutive Banach spaces (where this structure on F(T,E) is defined pointwise). In general F(T,E) is not an algebra for the pointwise multiplication, but for x,y, z ∈F(T,E) we characterize the case when x y=z.