Normality for an inclusion of ergodic discrete measured equivalence relations in the von Neumann algebraic framework. (English) Zbl 1136.46046

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 [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.


46L55 Noncommutative dynamical systems
28D99 Measure-theoretic ergodic theory


Zbl 1122.28012
Full Text: DOI