Abelian Von Neumann Algebra - Spatial Isomorphism

Spatial Isomorphism

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. This means that they cannot be extended to properly larger abelian algebras. They are also referred to as Maximal abelian self-adjoint algebras (or M.A.S.A.s). Another phrase used to describe them is abelian von Neumann algebras of uniform multiplicity 1; this description makes sense only in relation to multiplicity theory described below.

Von Neumann algebras A on H, B on K are spatially isomorphic (or unitarily isomorphic) if and only if there is a unitary operator U: H → K such that

In particular spatially isomorphic von Neumann algebras are algebraically isomorphic.

To describe the most general abelian von Neumann algebra on a separable Hilbert space H up to spatial isomorphism, we need to refer the direct integral decomposition of H. The details of this decomposition are discussed in decomposition of abelian von Neumann algebras. 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 {H_x}_{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.