In functional analysis, a branch of mathematics, an abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute.
The prototypical example of an abelian von Neumann algebra is the algebra L∞(X, μ) for μ a σ-finite measure on X realized as an algebra of operators on the Hilbert space L2(X, μ) as follows: Each f ∈ L∞(X, μ) is identified with the multiplication operator Of particular importance are the abelian von Neumann algebras on separable Hilbert spaces, particularly since they are completely classifiable by simple invariants.
Though there is a theory for von Neumann algebras on non-separable Hilbert spaces (and indeed much of the general theory still holds in that case) the theory is considerably simpler for algebras on separable spaces and most applications to other areas of mathematics or physics only use separable Hilbert spaces.
Although every standard measure space is isomorphic to one of the above and the list is exhaustive in this sense, there is a more canonical choice for the measure space in the case of abelian von Neumann algebras A: The set of all projectors is a
In the above theorem, the isomorphism is required to preserve the weak operator topology.
The complete classification of the operator algebra realizations of A is given by spectral multiplicity theory and requires the use of direct integrals.
Using direct integral theory, it can be shown that the abelian von Neumann algebras of the form L∞(X, μ) acting as operators on L2(X, μ) are all maximal abelian.
In particular: Theorem Any abelian von Neumann algebra on a separable Hilbert space H is spatially isomorphic to L∞(X, μ) acting on for some measurable family of Hilbert spaces {Hx}x ∈ X.
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 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.
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.