On the normalizing groupoids and the commensurability groupoids for inclusions of factors associated to ergodic equivalence relations-subrelations

摘要

It is shown that for the inclusion of factors (B subset of A) := (W*(S, omega) subset of W*(R, omega)) corresponding to an inclusion of ergodic discrete measured equivalence relations S subset of R, S is normal in R in the sense of Feldman-Sutherland-Zimmer [J. Feldman, CE. Sutherland, R.J. Zimmer, Subrelations of ergodic equivalence relations, Ergodic Theory Dynam. Systems 9 (1989) 239-269] if and only if A is generated by the normalizing groupoid of B. Moreover, we show that there exists the largest intermediate equivalence subrelation N-R(S) which contains S as a normal subrelation. We further give a definition of "commensurability groupoid" as a generalization of normality. We show that the commensurability groupoid of B in A generates A if and only if the inclusion B subset of A is discrete in the sense of Izumi-Longo-Popa [M. Izumi, R. Longo, S. Popa, A Galois correspondence for compact groups of automorphisms of von Neumann algebras with a generalization to Kac algebras, J. Funct. Anal. 155 (1998) 25-63]. We also show that there exists the largest equivalence subrelation Comm(R)(S) such that the inclusion B subset of W*(Comm(R)(S), omega) is discrete. It turns out that the intermediate equivalence subrelations N-R(S) and Comm(R)(S) subset of R thus defined can be viewed as groupoid-theoretic counterparts of a normalizer subgroup and a commensurability subgroup in group theory. (C) 2006 Elsevier Inc. All rights reserved.