Type I composition series - Imported Knowlpedia Content

Definition

A type I composition series for a $C^*$ -algebra $A$ is an ordinal-indexed increasing family $(I_\alpha)_{\alpha\leq\delta}$ of closed two-sided ideals such that $I_0=0$ , $I_\delta=A$ , and

I_\lambda=\overline{\bigcup_{\alpha<\lambda}I_\alpha}

at every limit ordinal $\lambda\leq\delta$ , while every successor inclusion is strict and $I_{\alpha+1}/I_\alpha$ is a continuous-trace $C^*$ -algebra. The term records a structural filtration, not a finite Jordan–Hölder series; both its ordinal length and its successive pieces can depend on choices.

Existence theorem

Every type I $C^*$ -algebra admits such an ascending series Dixmier, §10.5. The construction repeatedly passes to a quotient and selects a nonzero continuous-trace ideal, then takes closures at limit stages. The process ends because a strictly increasing chain of closed ideals cannot contain more members than the power set of the underlying algebra.

How the filtration is used

Properties stable under ideals, quotients, extensions, and transfinite inductive unions can often be proved for type I algebras by checking them on the continuous-trace layers. Each layer has a Hausdorff spectrum and is locally modeled by compact-operator algebras, so the series separates local geometric analysis from the possibly non-Hausdorff gluing of the full primitive ideal space.

Conventions and scope

Some sources use “GCR composition series” for a filtration whose layers are merely liminal, and then refine those layers further to continuous trace. The stronger continuous-trace convention is used here. The word “composition” does not assert that the layers are simple, unique, or of finite length.

References

Jacques Dixmier, C-Algebras, North-Holland Mathematical Library 15, North-Holland, 1977. Publisher record. Relevant: Chapter 4 and §10.5 on composition series and continuous-trace subquotients of type I algebras.
Gert K. Pedersen, C-Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: Chapter 6 on type I composition series.