Normality for an inclusion of ergodic discrete measured equivalence relations in the von Neumann algebraic framework (Q2468448)

It is shown that for the inclusion \((B\subseteq A):=(W^\star ({\mathcal S},\omega ))\subseteq W^\star ({\mathcal R},\omega))\) corresponding to an inclusion of ergodic discrete measured equivalence relations, \({\mathcal S}\subseteq {\mathcal R}\) is normal in \(\mathcal R\) in the sense of Feldman-Sutherland-Zimmer iff \(A\) is generated by the normalizing groupoid of \(B\). This fact has already been obtained in [\textit{H. Aoi} and the author, J. Funct. Anal. 240, 297--333 (2006; Zbl 1122.28012)]. The proof given there has an ``ergodic theory'' nature (partial Borel transformations, full groups, etc.). In the present paper, a new proof of the fact, in more operator-algebraic terms, is given. In connection with this, the normality in terms of minimal coactions of discrete groups is characterized, a notion of the normalizing groupoid for an inclusion of von Neumann algebras is introduced, and the normality is studied by using this normalizing groupoid.