Abelian Von Neumann Algebra - Classification

Classification

The relationship between commutative von Neumann algebras and measure spaces is analogous to that between commutative C*-algebras and locally compact Hausdorff spaces. Every commutative von Neumann algebra on a separable Hilbert space is isomorphic to L∞(X) for some standard measure space (X, μ) and conversely, for every standard measure space X, L∞(X) is a von Neumann algebra. This isomorphism as stated is an algebraic isomorphism. In fact we can state this more precisely as follows:

Theorem. Any abelian von Neumann algebra of operators on a separable Hilbert space is *-isomorphic to exactly one of the following

The isomorphism can be chosen to preserve the weak operator topology.

In the above list, the interval has Lebesgue measure and the sets {1, 2, ..., n} and N have counting measure. The unions are disjoint unions. This classification is essentially a variant of Maharam's classification theorem for separable measure algebras. The version of Maharam's classification theorem that is most useful involves a point realization of the equivalence, and is somewhat of a folk theorem.

Let μ and ν be non-atomic probability measures on standard Borel spaces X and Y respectively. Then there is a μ null subset N of X, a ν null subset M of Y and a Borel isomorphism

which carries μ into ν.

Notice that in the above result, it is necessary to clip away sets of measure zero to make the result work.

In the above theorem, the isomorphism is required to preserve the weak operator topology. As it turns out (and follows easily from the definitions), for algebras L∞(X, μ), the following topologies agree on norm bounded sets:

1. The weak operator topology on L∞(X, μ);
2. The ultraweak operator topology on L∞(X, μ);
3. The topology of weak* convergence on L∞(X, μ) considered as the dual space of L1(X, μ).

However, for an abelian von Neumann algebra A the realization of A as an algebra of operators on a separable Hilbert space is highly non-unique. The complete classification of the operator algebra realizations of A is given by spectral multiplicity theory and requires the use of direct integrals.