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

$$T \to {\rm{IK,}}\,\,\,\,\,\,\,\,\,t \mapsto \langle x_t ,a\rangle $$

is continuous. We define for every x ∈ F(T,E) an element $\tilde x$ ∈ C(T)$\bar \otimes$ E such that the map

$$f(T,E) \to b(T)\bar \otimes E,\,\,\,\,\,\,\,x \mapsto \tilde x$$

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 $\tilde x\tilde y = \tilde z$.