On $W^{\ast }$ embedding of $AW^{\ast }$-algebras

Abstract:

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