Normality for an inclusion of ergodic discrete measured equivalence relations in the von Neumann algebraic framework

Abstract

It is shown that for the inclusion of factors $(B\subseteq A):=(W^{*}(\mathit{S},\omega )\subseteq W^{*}(\mathit{R},\omega ))$ corresponding to an inclusion of ergodic discrete measured equivalence relations $\mathit{S}\subseteq \mathit{R}$, $\mathit{S}$ is normal in $\mathit{R}$ in the sense of Feldman-Sutherland-Zimmer ([9]) if and only if $A$ is generated by the normalizing groupoid of $B$. Though this fact has been already obtained in [3], we reprove it here by a quite different method.