Abelian von Neumann algebra

In functional analysis, 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. Note that if the measure spaces (X, μ) is a standard measure space (that is X − N is a standard Borel space for some null set N and μ is a σ-finite measure) then L²(X, μ) is separable.

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

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