Commutative von Neumann algebra - Imported Knowlpedia Content

Definition

A commutative von Neumann algebra is a von Neumann algebra $M$ in which $xy=yx$ for all $x,y\in M$ . Equivalently, $M$ equals its center. The fundamental concrete model is $L^\infty(X,\mu)$ , acting on $L^2(X,\mu)$ by multiplication:

(M_f\xi)(x)=f(x)\xi(x).

For suitable localizable measure spaces, every commutative von Neumann algebra is isomorphic as a von Neumann algebra to such an $L^\infty$ algebra. The measure model is determined by the associated measure algebra, not by a preferred point-set presentation of $X$ .

Projections and measure

The projections in $L^\infty(X,\mu)$ are the equivalence classes of indicator functions $1_E$ of measurable sets, where sets differing by a null set define the same projection. Joins, meets, and complements of projections correspond to the Boolean operations on measurable sets modulo null sets. This complete projection lattice is the measure-theoretic skeleton of the algebra.

Atomic and diffuse parts

A commutative von Neumann algebra is atomic when every nonzero projection dominates a minimal projection; typical examples are $\ell^\infty(I)$ . It is diffuse when it has no nonzero minimal projections, as for $L^\infty([0,1])$ with Lebesgue measure. In general the algebra decomposes canonically into central atomic and diffuse summands.

Contrast with commutative C*-algebras

A commutative $C^*$ -algebra is described by continuous functions on a locally compact space, whereas a commutative von Neumann algebra is governed by essentially bounded measurable functions and is monotone complete. Its natural topology is the weak-star topology coming from its predual, not the uniform topology alone. This weak-star topology and the extra order structure record measure classes and normal integration.

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 2002. Publisher record. Relevant: Chapter III, §1 on commutative von Neumann algebras and measure representations.
Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, Volume II: Advanced Theory, American Mathematical Society, 1997. Publisher record. Relevant: the representation and projection theory of abelian von Neumann algebras.