Type I stability under ideals and quotients - Imported Knowlpedia Content

Statement

Let $A$ be a type I $C^*$ -algebra and let $I$ be a closed two-sided ideal. Then both $I$ and $A/I$ are type I $C^*$ -algebras. Thus in every short exact sequence

0\longrightarrow I\longrightarrow A\longrightarrow A/I\longrightarrow0

with type I middle algebra, the ideal and quotient inherit the type I property. Equivalently, the class of GCR algebras is closed under passing to closed ideals and quotients. The result applies without separability or unitality assumptions Pedersen, Proposition 6.2.6.

Representation-theoretic mechanism

An irreducible representation of $A/I$ lifts along the quotient map to an irreducible representation of $A$ , so its image contains the compact operators. For an irreducible representation of $I$ , the standard extension to $A$ transfers the same compact-operator containment. These two correspondences explain why both halves of the theorem are representation-theoretically natural.

Three-space form

Type I is also an extension-stable property: if $I$ and $A/I$ are type I, then $A$ is type I. Consequently, in a short exact sequence, knowing any appropriate ideal–quotient decomposition permits induction through a composition series Pedersen, Proposition 6.2.6. This converse is not automatic for an arbitrary class of $C^*$ -algebras.

Example

The compact operators $K(H)$ form an ideal in $K(H)+\mathbb C I_H$ , and the quotient is $\mathbb C$ . Both are type I, so extension stability recovers that the unitized algebra is type I. The example also shows that the theorem does not preserve the stronger liminal property through arbitrary extensions.

References

Gert K. Pedersen, C-Algebras and Their Automorphism Groups, Academic Press, 1979. DOI record for the revised edition. Relevant: Proposition 6.2.6 on ideals, quotients, and extensions of type I algebras.
Jacques Dixmier, C-Algebras, North-Holland Mathematical Library 15, North-Holland, 1977. Publisher record. Relevant: Chapter 4 on postliminal ideals and quotients.