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

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.