There are several approaches to the topology on the spectrum. The approaches presented here are that of:

- Ideal-theoretic approach: the topology is lifted from the hull-kernel topology on the primitive ideal space.
- Viewing the spectrum as the image under the GNS-construction of a collection of positive functionals and pushing the weak-$\ast$ topology forward.

## Definitions: Hull-kernel topology

Recall that *primitive ideals* are those ideals that are the kernels of irreducible representations. These are

- closed, since representations of $C^\ast$-algebras are automatically closed;
- every closed ideal $I$ in a $C^\ast$-algebra $A$ is the intersection of the primitive ideals containing it;
- if $I$ is a primitive ideal in $A$ and $J,K$ are (closed) ideals s.t. $J\cap K\subset I$, then either $K\subset I$ or $J\subset I$,
*viz.*every primitive ideal is prime; - For separable $C^\ast$-algebras any closed prime ideal is also primitive.

*primitive ideal set*$\operatorname{Prim}A$ of a $C^\ast$-algebra $A$ is the set of primitive ideals in $A$.

Usually the primitive ideal set is understood to be equipped with the *Jacobson topology*, also called *hull-kernel topology*. The topology is defined by let the closure $\overline{F}$ of $F$ to be
$$
\overline{F} \colon = \{P\in\operatorname{Prim}A|\bigcap_{I\in F} I\subset P\}.
$$
the set of all the primitive ideals containing the intersection of elements in $F$. The topology on $\operatorname{Prim}A$ is defined by taking all the subsets $F$ s.t. $F=\overline{F}$ to be the closed sets.

*spectrum set*$\hat{A}=\operatorname{Spec}A$ of a $C^\ast$-algebra $A$ is the set of unitary equivalence classes of irreducible representations of $A$.

Since the kernels of equivalent representations are the same, there is a canonical surjection $\pi\mapsto \operatorname{Ker}\pi$ of $\hat{A}$ onto $\operatorname{Prim}A$.

- For many $C^\ast$-algebras this map is also injective,
*e.g.*the algebra of functions over a compact Hausdorff space. - The map should really be understood as taking an equivalence class $[\pi]$ to its kernel, which are equal for all the elements in the equivalence class.

The surjection $\hat{A} \to \operatorname{Prim}A:\pi\mapsto\operatorname{Ker}\pi$ allows the hull-kernel topology on the primitive ideal set to be pulled back to $\hat{A}$ and endow $\hat{A}$ with this topology. Thus,

- The
*primitive ideal space*$\operatorname{Prim}A$ of a $C^\ast$-algebra $A$ is the set of primitive ideals in $A$ endowed with the hull-kernel topology. - The
*spectrum*$\hat{A}=\operatorname{Spec}A$ of a $C^\ast$-algebra $A$ is the set of unitary equivalence classes of irreducible representations of $A$ with the topology induced from $\operatorname{Prim}A$: $S\subset \hat{A}$ is open iff $\{\operatorname{Ker}\pi|\pi\in S\}$ is open in $\operatorname{Prim}A$.

## Definition: GNS-construction

Let $P(A)$ denote the space of pure states of a $C^\ast$-algebra $A$, endowed with the weak-$\ast$ topology.

Thus the weak-$\ast$ topology on $P(A)$ descends to a quotient topology on $\hat{A}$.