Type I factor representation - Imported Knowlpedia Content

Definition

Let $A$ be a $C^*$ -algebra. A type I factor representation of $A$ is a nondegenerate $*$ -representation $\pi\colon A\to B(H)$ such that the generated von Neumann algebra

\pi(A)''

is a type I factor. Here the double prime denotes the bicommutant in $B(H)$ . Factoriality means that the center of $\pi(A)''$ consists only of scalar multiples of the identity. Both requirements matter: a type I generated algebra can have nontrivial center, while a factor can have type II or type III.

Spatial form

A type I factor acting on $H$ has a tensor-product decomposition

H\cong K\otimes L,\qquad \pi(A)''\cong B(K)\otimes 1_L.

The auxiliary Hilbert space $L$ records multiplicity. Thus the generated algebra is abstractly isomorphic to $B(K)$ , but need not equal all of $B(H)$ . When $L$ is one-dimensional, $\pi$ is irreducible.

Role in type I C*-algebras

A $C^*$ -algebra is type I exactly when every factor representation is of type I. This factor-representation criterion is equivalent to the compact-operator criterion in the definition of a type I $C^*$ -algebra. It is especially useful for representations that are not irreducible but still generate a factor.

Terminology

Some authors call any representation $\pi$ with $\pi(A)''$ a type I von Neumann algebra a type I representation, without requiring the generated algebra to be a factor. Such a representation may decompose over a nontrivial center. “Type I factor representation” includes the factoriality hypothesis and avoids this ambiguity.

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter V on type I von Neumann algebras and representations.
Gert K. Pedersen, C-Algebras and Their Automorphism Groups*, 2nd ed., Academic Press, 2018. DOI record. Relevant: Chapter 6 on factor representations and type I $C^*$ -algebras.