کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
4592237 | 1630609 | 2006 | 37 صفحه PDF | دانلود رایگان |
![عکس صفحه اول مقاله: On the normalizing groupoids and the commensurability groupoids for inclusions of factors associated to ergodic equivalence relations–subrelations On the normalizing groupoids and the commensurability groupoids for inclusions of factors associated to ergodic equivalence relations–subrelations](/preview/png/4592237.png)
It is shown that for the inclusion of factors (B⊆A):=(W∗(S,ω)⊆W∗(R,ω)) corresponding to an inclusion of ergodic discrete measured equivalence relations S⊆R, S is normal in R in the sense of Feldman–Sutherland–Zimmer [J. Feldman, C.E. 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 NR(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⊆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 CommR(S) such that the inclusion B⊆W∗(CommR(S),ω) is discrete. It turns out that the intermediate equivalence subrelations NR(S) and CommR(S)⊆R thus defined can be viewed as groupoid-theoretic counterparts of a normalizer subgroup and a commensurability subgroup in group theory.
Journal: Journal of Functional Analysis - Volume 240, Issue 2, 15 November 2006, Pages 297-333