Abelian Von Neumann Algebra - Point and Spatial Realization of Automorphisms

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

1. 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;
2. 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 {H_x}_{x ∈ X}, {K_y}_{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 {U_x}_{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.