Glimm's type I theorem

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.

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.

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.