H is spatially isomorphic to L∞(X, μ) acting on for some measurable family of Hilbert spaces {Hx}x ∈ X. Note that for abelian von Neumann algebras acting on such direct integral spaces, the equivalence of the weak operator topology, the ultraweak topology and the weak* topology on norm bounded sets still hold. Point and spatial realization of automorphisms Many problems in ergodic theory reduce to problems about automorphisms of abelian von Neumann algebras. In that regard, the following results are useful: Theorem. Suppose μ, ν are standard measures on X, Y respectively. Then any involutive isomorphism which is weak*-bicontinuous corresponds to a point transformation in the following sense: There are Borel null subsets M of X and N of Y and a Borel isomorphism such that η carries the measure μ into a measure μ' on Y which is equivalent to ν in the sense that μ' and ν have the same sets of measure zero; η realizes the transformation Φ, that is Note that in general we cannot expect η to carry μ into ν. The next result concerns unitary transformations which induce a weak*-bicontinuous isomorphism between abelian von Neumann algebras. Theorem. Suppose μ, ν are standard measures on X, Y and for measurable families of Hilbert spaces {Hx}x ∈ X, {Ky}y ∈ Y. If U : H → K is a unitary such that then there is an almost everywhere defined Borel point transformation η : X → Y as in the previous theorem and a measurable family {Ux}x ∈ X of unitary operators such that where the expression in square root sign is the Radon–Nikodym derivative of μ η−1 with respect to ν. The statement follows combining the theorem on point realization of automorphisms stated above with the theorem characterizing the algebra of diagonalizable operators stated in the article on direct integrals. Notes References J. Dixmier, Les algèbres d'opérateurs dans l'espace Hilbertien, Gauthier-Villars, 1969. See chapter I, section 6. Masamichi Takesaki ''Theory of Operator Algebras I,II,III", encyclopedia of mathematical sciences, Springer-Verlag, 2001–2003 (the first volume was published 1979 in 1. Edition) Von Neumann algebras