On 𝑊* embedding of 𝐴𝑊*-algebras

Abstract

An A W ∗ A{W^ \ast } -algebra N with a separating family of completely additive states and with a family { e α : α ∈ A } \{ {e_\alpha }:\alpha \in A\} of mutually orthogonal projections such that lub α e α = 1 {\operatorname {lub} _\alpha }{e_\alpha } = 1 and e α N e α {e_\alpha }N{e_\alpha } is a W ∗ {W^ \ast } -algebra for each α ∈ A \alpha \in A is shown to have a faithful representation as a ring of operators. This gives a new and considerably shorter proof that a semifinite A W ∗ A{W^ \ast } -algebra with a separating family of completely additive states has a faithful representation as a ring of operators.