Maximal abelian subalgebras of von Neumann algebras and representations of equivalence relations
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- by Colin E. Sutherland PDF
- Trans. Amer. Math. Soc. 280 (1983), 321-337 Request permission
Abstract:
We associate to each pair $(\mathcal {M},\mathcal {A})$, (with $\mathcal {M}$ a von Neumann algebra, and $\mathcal {A}$ a maximal abelian subalgebra) a representation $\alpha$ of the Takesaki equivalence relation $\mathcal {R} (\mathcal {M},\mathcal {A})$ of $(\mathcal {M},\mathcal {A})$ as automorphisms of a ${{\text {I}}_\infty }$ factor. Conversely each such representation $\alpha$ of $\mathcal {R}$ on $(X,\mu )$ as automorphisms of $\mathcal {B} (\mathcal {H})$ determines a von Neumann algebra-abelian subalgebra pair $S’ (\mathcal {R},\alpha ) = (\mathcal {N},\mathcal {B})$ where $\mathcal {N}$ is the commutant of the algebra of "self-intertwiners" for $\alpha$ and $\mathcal {B} = {L^\infty }(X,\mu ) \otimes 1$ on ${L^2}(X,\mu ) \otimes \mathcal {H}$. The main concern is the assignments $(\mathcal {M},\mathcal {A}) \to \mathcal {T}\;(\mathcal {M},\mathcal {A}) = (\mathcal {R} (\mathcal {M},\mathcal {A}),\alpha )$ and $(\mathcal {R},\alpha ) \to S’(\mathcal {R},\alpha )$, and in particular, the extent to which they are inverse to each other—this occurs if $\mathcal {R}$ is countable nonsingular and $\alpha$ is (conjugation by) a projective square-integrable representation (cf. [8]), or if $\mathcal {A}$ is a Cartan subalgebra (cf. [5]), among other cases. A partial dictionary between the representations $(\mathcal {R},\alpha )$ and pairs $(\mathcal {M},\mathcal {A})$ is given—thus if $\mathcal {R}$ is countable nonsingular and $\alpha$ is what we term replete, $S’(\mathcal {R},\alpha )$ is injective whenever $\mathcal {R}$ is amenable, and a complete Galois theory generalizing that for crossed products by discrete groups is available. We also show how to construct various pathological examples such as a singular maximal abelian subalgebra $\mathcal {A} \subseteq \mathcal {M}$ for which the Takesaki equivalence relation $\mathcal {R} (\mathcal {M},\mathcal {A})$ is nontrivial.References
- Alain Connes, Une classification des facteurs de type $\textrm {III}$, Ann. Sci. École Norm. Sup. (4) 6 (1973), 133–252 (French). MR 341115 —, Sur la théorie non-commutative de l’intégration, Lecture Notes in Math., vol. 725, Springer-Verlag, Berlin and New York.
- H. A. Dye, On groups of measure preserving transformations. I, Amer. J. Math. 81 (1959), 119–159. MR 131516, DOI 10.2307/2372852
- Jacob Feldman and Calvin C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Amer. Math. Soc. 234 (1977), no. 2, 289–324. MR 578656, DOI 10.1090/S0002-9947-1977-0578656-4
- Jacob Feldman and Calvin C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Amer. Math. Soc. 234 (1977), no. 2, 289–324. MR 578656, DOI 10.1090/S0002-9947-1977-0578656-4
- Peter Hahn, Haar measure for measure groupoids, Trans. Amer. Math. Soc. 242 (1978), 1–33. MR 496796, DOI 10.1090/S0002-9947-1978-0496796-6
- Peter Hahn, The regular representations of measure groupoids, Trans. Amer. Math. Soc. 242 (1978), 35–72. MR 496797, DOI 10.1090/S0002-9947-1978-0496797-8
- Peter Hahn, Reconstruction of a factor from measures on Takesaki’s unitary equivalence relation, J. Functional Analysis 31 (1979), no. 3, 263–271. MR 531129, DOI 10.1016/0022-1236(79)90001-6
- A. A. Kirillov, Elements of the theory of representations, Grundlehren der Mathematischen Wissenschaften, Band 220, Springer-Verlag, Berlin-New York, 1976. Translated from the Russian by Edwin Hewitt. MR 0412321
- George W. Mackey, Ergodicity in the theory of group representation, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 401–405. MR 0435287
- George W. Mackey, Borel structure in groups and their duals, Trans. Amer. Math. Soc. 85 (1957), 134–165. MR 89999, DOI 10.1090/S0002-9947-1957-0089999-2
- George W. Mackey, Point realizations of transformation groups, Illinois J. Math. 6 (1962), 327–335. MR 143874
- Colin E. Sutherland, Cohomology and extensions of von Neumann algebras. I, II, Publ. Res. Inst. Math. Sci. 16 (1980), no. 1, 105–133, 135–174. MR 574031, DOI 10.2977/prims/1195187501
- Masamichi Takesaki, On the unitary equivalence among the components of decompositions of representations of involutive Banach algebras and the associated diagonal algebras, Tohoku Math. J. (2) 15 (1963), 365–393. MR 164249, DOI 10.2748/tmj/1178243773
- Masamichi Takesaki, Duality for crossed products and the structure of von Neumann algebras of type III, Acta Math. 131 (1973), 249–310. MR 438149, DOI 10.1007/BF02392041
- Yoshiomi Nakagami and Masamichi Takesaki, Duality for crossed products of von Neumann algebras, Lecture Notes in Mathematics, vol. 731, Springer, Berlin, 1979. MR 546058
- Robert J. Zimmer, Hyperfinite factors and amenable ergodic actions, Invent. Math. 41 (1977), no. 1, 23–31. MR 470692, DOI 10.1007/BF01390162
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 280 (1983), 321-337
- MSC: Primary 46L10
- DOI: https://doi.org/10.1090/S0002-9947-1983-0712263-8
- MathSciNet review: 712263