Reconstruction of a factor from measures on Takesaki's unitary equivalence relation

AbstractLet Ol≅L∞(S, μ) be a maximal abelian subalgebra of the factor F on separable Hilbert space with modular involution J. (Ol∪ JOlJ)″ is represented naturally as L∞(S × S, λ). If Takesaki's unitary equivalence relation R ⊂ S × S is not λ-null, it is a measure groupoid. If it is conull, and (Ol∪ JOlJ)″ is maximal abelian, F and Ol are reconstructed by the σ-left regular representation procedure. Examples show that these hypotheses are not always satisfied. An application shows that the L∞ spectrum of a properly infinite ergodic transformation is null with respect to the L2 spectrum