Last Thursday, I gave a talk on commutative von Neumann algebras. I revised my notes slightly and filled in some gaps in my presentation. All the material comes from G. Pedersen’s book *C*-algebras and their Automorphism Groups*. Thanks go out to Eusebio Gardella and Michael Sun for helping me fill in my ignorance and seeing the argument more fully.

A von Neumann algebra is a weak operator closed, self-adjoint subalgebra of bounded linear operators on a Hilbert space. For the sake of convenience, we will assume that every von Neumann algebra contains the identity operator, thus forming a unital algebra. In addition and for more than convenience, we will be assuming that our Hilbert spaces are separable.

Clearly the space of all bounded linear operators forms a von Neumann algebra and will be denoted . Immediately, we notice in contrast to the -algebra case that the definition of von Neumann algebra we provided is distinctly an operator algebra. Nonetheless, we have a characterization of commutative von Neumann algebras in terms of function spaces.

Let be a compact, Hausdorff space and a probability Borel measure. We can embed into by considering functions as multiplication operators, i.e. is identified with the operator . This embedding is in fact, an isometric homomorphism and the image is weak-operator closed, thereby forming a von Neumann algebra. From now on, we will be making this identification implicitly and simply say that is a von Neumann algebra.

The purpose of this post is to provide a proof of the fact that every commutative von Neumann algebra (on a separable Hilbert space) is isomorphic to for appropriate choice of and . But first we need to make two observations about .

First, is maximal in the sense that there is no commutative subalgebra that properly contains in . And second, the weak closure of identified as multiplication operators is . For good reasons that won’t be discussed, we denote the weak operator closure of a subalgebra to be , and therefore .

We will first prove the theorem for maximal commutative von Neumann algebras, and then I will subsequently handwave the general case, since this is what I did during the talk. The benefit of having a maximal commutative algebra is the existence of cyclic vectors. A cyclic vector of an algebra (or even subset) of is a vector such that the set is dense in . The existence of cyclic vectors guarantee that certain isomorphisms of concrete -algebras are in fact unitary equivalences and that therefore isomorphisms of -algebras lift to their weak operator closures and extend to isomorphisms of von Neumann algebras.

**Theorem.** Let and be isomorphic -algebras with cyclic vectors and , respectively and isomorphism . If for all , then there exists a unitary such that and .

First we define by . This map is clearly linear (provided that it is well-defined). Note that

. So is a well-defined unitary, and therefore continuous.

Thus extends to a map , which is unitary. Finally, . So .

**Corollary.** If and be isomorphic -algebras with cyclic vectors and and isomorphism , then is isomorphic to provided .

This follows from the fact that conjugation by a unitary is weakly continuous. Now we turn our attention back to maximal commutative von Neumann algebras.

**Theorem.** If is separable and is a maximal commutative von Neumann algebra, then has a cyclic vector.

By Zorn’s lemma, there exists a maximal set of unit vectors whose projections onto the closure of are mutually orthogonal. This is countable by separability, since maximal sets of orthonormal vectors are countable. The span of the spaces is , since otherwise there would be a unit vector in the orthogonal complement and it can be seen that would be in the orthogonal complement, contradicting maximality.

Set . Since . And therefore is a cyclic vector of .

**Theorem.** Every maximal commutative von Neumann algebra on is unitarily equivalent to for some compact, metrizable space and probability measure .

Since the unit ball of is weakly compact, metrizable, so is the unit ball of and thus separable. Take a -subalgebra generated by a countable weakly dense subset of unit vectors and call it .

By Gelfand’s representation theorem, for some compact, metrizable space . Furthermore, the map defines a linear functional on , where is a cyclic vector of . By Riesz representation theorem, there is a positive Borel measure such that Note that . Notice that if is a cyclic vector of , then $v_1$ is a cyclic vector of and also the constant function is a cyclic vector of , since is dense in . So by our corollary, we see that the isomorphism between and lifts to an isomorphism between and .

**Corollary.** Every commutative von Neumann algebra on is unitarily equivalent to for some compact, metrizable space and probability measure .

Unfortunately, I didn’t get to the proof of this. Nor given the above can I prove it now. The basic idea is take our von Neumann algebra and prove that it is isomorphic to a corner, where it has a new representation where our algebra is maximal and then apply the previous theorem.