Aoi, Hisashi; Yamanouchi, Takehiko On the normalizing groupoids and the commensurability groupoids for inclusions of factors associated to ergodic equivalence relations-subrelations. (English) Zbl 1122.28012 J. Funct. Anal. 240, No. 2, 297-333 (2006). Consider a discrete measured equivalence relation \(\mathcal{R}\) on a standard probability space \((X,\Sigma,\mu),\) which is quasi-invariant with respect to \(\mu,\) and a normalized Borel 2-cocycle \(\omega\) from \(\mathcal{R}\) into \(S^{1}.\) A classical construction due to J. Feldman and C. C. Moore [Trans. Am. Math. Soc. 234, 325–359 (1977; Zbl 0369.22010)] allows us to associate to \(\mathcal{R}\) and \(\omega\) a von Neumann \(W^{\star}\left( \mathcal{R},\omega\right) .\) Theorem 5.16 shows that an ergodic equivalence subrelation \(\mathcal{S}\) is normal in \(\mathcal{R}\) if and only if the corresponding \(W^{\star}\left( \mathcal{R},\omega\right) \) is generated by the normalizing grupoid of \(W^{\star}\left( \mathcal{S},\omega\right) .\) As a consequence, for every ergodic subrelation \(\mathcal{S}\) of \(\mathcal{R}\) there always exists the largest intermediate equivalence subrelation which contains \(\mathcal{S}\) as a normal subrelation.Then the authors introduce a concept of commensurability groupoid as a generalization of normality and prove that the commensurability grupoid \(C\mathcal{G}(B),\) of \(B=W^{\star }\left( \mathcal{S},\omega\right) \) in \(A=W^{\star}\left( \mathcal{R} ,\omega\right) ,\) generates \(A\) if and only if the inclusion \(B\subseteq A\) is discrete in the sense of M. Izumi, R. Longo and S. Popa [J. Funct. Anal. 155, No. 1, 25–63 (1998; Zbl 0915.46051)]. In fact, Theorem 7.10 states that the inclusion \(B\subseteq C\mathcal{G}(B)^{\prime\prime}\) is discrete and \(C\mathcal{G} (B)^{\prime\prime}\) is the largest among the intermediate subfactors \(M\) of \(B\subseteq A\) such that \(B\subseteq M\) is discrete. Reviewer: Constantin Niculescu (Craiova) Cited in 1 ReviewCited in 3 Documents MSC: 28D99 Measure-theoretic ergodic theory 46L35 Classifications of \(C^*\)-algebras Keywords:von Neumann algebra; discrete inclusion Citations:Zbl 0369.22010; Zbl 0915.46051 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Aoi, H., A construction of equivalence subrelations for intermediate subalgebras, J. Math. Soc. Japan, 55, 713-725 (2003) · Zbl 1033.46046 [2] H. Aoi, T. Yamanouchi, A characterization of coactions whose fixed-point algebras contain special maximal abelian ∗-subalgebras, Ergodic Theory Dynam. Systems, in press; H. Aoi, T. 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