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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Maximal abelian subalgebras of von Neumann algebras and representations of equivalence relations

Author: Colin E. Sutherland
Journal: Trans. Amer. Math. Soc. 280 (1983), 321-337
MSC: Primary 46L10
MathSciNet review: 712263
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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^{\prime}\,(\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^{\prime}(\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^{\prime}(\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.

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Keywords: Maximal abelian subalgebras, Cartan subalgebra, groupoid cohomology, transverse measure
Article copyright: © Copyright 1983 American Mathematical Society