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⊆A):=(W^*(\\mathcal{S},ω)⊆W^*(\\mathcal{R},ω)) corresponding to an inclusion of ergodic discrete measured equivalence relations \\mathcal{S}⊆\\mathcal{R}, \\mathcal{S} is normal in \\mathcal{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.