Von Neumann algebra - Imported Knowlpedia Content

Definition

Let $H$ be a Hilbert space. A von Neumann algebra on $H$ is a unital $*$ -subalgebra $M\subseteq B(H)$ , the algebra of bounded operators, that is closed in the weak operator topology. Equivalently, $M$ is closed in the strong operator topology or satisfies

M=M'',

where $M''$ is its bicommutant. An abstract $W^*$ -algebra is a $C^*$ -algebra that is the dual Banach space of a Banach-space predual. Every abstract $W^*$ -algebra has a faithful representation as a concrete von Neumann algebra.

Closure and the bicommutant theorem

For a unital self-adjoint subalgebra $A\subseteq B(H)$ , the von Neumann bicommutant theorem identifies its weak-operator closure, strong-operator closure, and double commutant $A''$ . The hypotheses matter: an arbitrary weakly closed nonself-adjoint operator algebra is not a von Neumann algebra. Norm closure alone produces only a $C^*$ -algebra and is generally smaller Takesaki, Chapter III.

The predual and normality

The predual $M_*$ of a von Neumann algebra is unique up to isometric isomorphism. Its elements are the normal bounded linear functionals, and the weak-star topology $\sigma(M,M_*)$ agrees with the ultraweak topology in a concrete faithful representation. This predual is extra analytic structure not carried by an arbitrary $C^*$ -algebra; it supports normal states, normal maps, and monotone convergence.

Examples and scope

The full algebra $B(H)$ , every commutant $S'\subseteq B(H)$ , and $L^\infty(X,\mu)$ acting by multiplication on $L^2(X,\mu)$ are von Neumann algebras. A norm-closed algebra such as $C([0,1])$ in its multiplication representation is usually not weakly closed. Calling a $W^*$ -algebra a von Neumann algebra suppresses the choice of faithful concrete representation but is standard when no confusion can result.

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter III on von Neumann algebras, bicommutants, preduals, and normal functionals.
Shôichirô Sakai, $C^*$ -Algebras and $W^*$ -Algebras, Springer, 1971; Classics in Mathematics reprint, 1998. Publisher record. Relevant: Chapter 1 on the abstract predual characterization.