Von Neumann algebra generated by a representation

Definition

Let $\pi:A\to\mathcal B(H)$ be a representation of a $C^*$ -algebra. The von Neumann algebra generated by $\pi$ is

W^*(\pi(A))=\pi(A)''.

By the von Neumann bicommutant theorem, this is the weak-operator and strong-operator closure of the unital $*$ -algebra generated by $\pi(A)$ . It is the smallest von Neumann subalgebra of $\mathcal B(H)$ containing the represented image. If $\pi$ is nondegenerate, the identity already lies in the strong closure of $\pi(A)$ , so $W^*(\pi(A))$ is the weak or strong closure of $\pi(A)$ itself.

Why the bicommutant is used

Taking a bicommutant automatically adjoins the identity and closes in the relevant operator topologies. This matters for a degenerate representation: the weak closure of $\pi(A)$ can act as zero on a complementary subspace and therefore need not contain $I_H$ , while $\pi(A)''$ is unital. Adding $I_H$ before taking the weak closure gives the same algebra as the bicommutant Takesaki, Chapter III, §2.

Dependence on the representation

The construction depends on the representation, not only on the abstract algebra $A$ . Different representations of the same $C^*$ -algebra may generate nonisomorphic von Neumann algebras or different normal representation theories. The universal representation produces the enveloping von Neumann algebra $A^{**}$ , whereas a particular quotient or factor representation can generate a much smaller algebra.

Role in representation theory

Properties formulated using weak limits become visible only after passing from $\pi(A)$ to $W^*(\pi(A))$ . A representation is factorial when this generated algebra is a factor. Quasi-equivalence of representations asks whether their generated von Neumann algebras are normally isomorphic in a way that agrees on the common $C^*$ -algebra.

References

Jacques Dixmier, C-Algebras*, North-Holland, 1977. Publisher record. Relevant: Chapter 12 on the enveloping von Neumann algebra of a $C^*$ -algebra.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 2002. Publisher record. Relevant: Chapter III, §2 on representations and generated von Neumann algebras.