Glimm's type I theorem - Imported Knowlpedia Content

Statement

Let $A$ be a separable $C^*$ -algebra and let $\widehat A$ be its unitary dual with the Fell topology. Glimm's theorem says that the following are equivalent:

$A$ is a type I $C^*$ -algebra;
the kernel map from $\widehat A$ to the

primitive ideal space $\operatorname{Prim}(A)$ is bijective;

$\widehat A$ is a $T_0$ space; and
every nonempty closed subset of $\widehat A$ contains a dense relatively

open Hausdorff subset.

Thus type I is exactly the regime in which irreducible representations are topologically separated well enough to admit a tractable classification Glimm, main theorem.

Meaning of the kernel condition

The kernel map is always surjective. Its injectivity says that two irreducible representations with the same primitive ideal are unitarily equivalent. Because $\operatorname{Prim}(A)$ is always $T_0$ , failure of the theorem's conditions measures representation-theoretic ambiguity that the primitive ideal alone cannot resolve.

Almost Hausdorff structure

Condition 4 is often summarized by saying that $\widehat A$ is almost Hausdorff. It supplies Hausdorff pieces densely inside every closed representation-theoretic stratum and underlies the construction of transfinite composition series with continuous-trace layers Dixmier, Chapter 4.

Conventions and scope

The name “Glimm dichotomy” is also used for stronger descriptive-set-theoretic consequences on the non-type-I side. Those refinements are not included in the statement above.

References

James G. Glimm, “Type I $C^*$ -algebras,” Annals of Mathematics 73 (1961), 572–612. DOI record. Relevant: the main characterization of separable type I algebras.
Jacques Dixmier, C-Algebras, North-Holland Mathematical Library 15, North-Holland, 1977. Publisher record. Relevant: Chapter 4 on type I spectra and almost-Hausdorff structure.