Type I C*-algebra - Imported Knowlpedia Content

Definition

A $C^*$ -algebra $A$ is a type I $C^*$ -algebra, also called GCR or postliminal, if for every irreducible representation $\pi\colon A\to B(H_\pi)$ , the represented algebra contains the compact operators:

K(H_\pi)\subseteq\pi(A).

This condition applies to nonunital as well as unital algebras and is independent of the chosen representative of the unitary-equivalence class. It is the $C^*$ -algebraic type I condition, distinct from saying that a particular von Neumann algebra is type I.

Equivalent viewpoints

Equivalently, every factor representation $\pi$ of $A$ generates a type I von Neumann algebra $\pi(A)''$ . Another characteristic consequence is that an irreducible representation is determined up to unitary equivalence by its kernel. Hence the spectrum of $A$ maps bijectively to its primitive ideal space Pedersen, Chapter 6.

Examples and contrasts

Every commutative $C^*$ -algebra and every compact-operator algebra $K(H)$ is type I. The stronger CCR, or liminal, condition requires $\pi(A)=K(H_\pi)$ for every irreducible $\pi$ ; type I only requires containment. Group $C^*$ -algebras of type I groups supply important noncommutative examples, while many discrete groups have non-type-I full group algebras.

References

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