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@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!

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@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 a countably infinite dimensional Hilbert space).
• $Prim(A)$ 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!