@Meyer has provided a very good answer, so I just want to add something from my notes from Professor Rieffel's $C^\ast$-algebra class about the primitive ideal space, which is a canonical topological space associated to a $C^\ast$-algebra (in fact, it is associated to any normed $*$-algebra, but let's keep it "easy"...).
Let $A$ be a $C^\ast$-algebra.
Definition: An ideal $I\subset A$ is called primitive if it is the kernel of an irreducible representation. Denote the set of all primitive ideals by $Prim(A)$.
Note that $Prim(A)$ is always a set as it is a subset of $P(A)$, the power set of $A$. An important theorem:
If $A$ is GCR, then the map $\widehat{A}\to Prim(A)$ by $(\pi, H)\mapsto \ker(\pi)$ is a bijection (where $\widehat{A}$ is the "set" of equivalence classes of irreducible representations).
This is far from the case if $A$ is NCR (there is some set theory here, such as the term "unclassifiable," that I don't want to get into as I am not a set theorist), but $Prim(A)$ is still a set, so we still get a topological space using the following fact:
All primitive ideals are prime.
The space of prime ideals of $A$ comes with the Jacobson, or "hull-kernel" topology, so we get the relative topology on $Prim(A)$.
A few facts:
- In general, $Prim(A)$ is not Hausdorff, but it is $T_0$ (look at $B_0(H)$, the compact operators on see @Meyer's comment for a countably infinite dimensional Hilbert space)counterexample).
- $Prim(A)$ is locally compact.
- If $A$ is separable, $Prim(A)$ has the Baire Category Property.
Once again, this is all from a course I took from Professor Rieffel. I hope it helps!
