An extension of Kakutani’s theorem on infinite product measures to the tensor product of semifinite 𝑤*-algebras

Suppose that (s%)if, is a family of semifinite w*-algebras and that /( is a normal state of sft with /(1) = 1 for each i el. Let s/=0ls, (s/u /) and let AK-+Ax denote the natural injection of ^ into si. (The notation is explained in 3 below; si is the (/^-incomplete direct product of (s): see [9], [12] or [1].) Given a normal state vt of s/t for each i e I, a normal state v of s/ is written (x)je/ vt when " in ) = n "(^o for all Aiesii and all finite subsets F of /.