Spectrum and Primitive Ideal Space

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

  1. Ideal-theoretic approach: the topology is lifted from the hull-kernel topology on the primitive ideal space.
  2. 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

  1. closed, since representations of $C^\ast$-algebras are automatically closed;
  2. every closed ideal $I$ in a $C^\ast$-algebra $A$ is the intersection of the primitive ideals containing it;
  3. 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;
  4. For separable $C^\ast$-algebras any closed prime ideal is also primitive.
Definition(Primitive ideal set) . The 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.

Definition(Spectrum set) . The 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 $\operatorname{Prim}A$.

Remark.
  • 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,

Definition(Spectrum and primitive ideal space) .
  • 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.

Theorem. The GNS-map $\Lambda:P(A)\to \hat{A}:\rho\mapsto \pi_\rho$ is a continuous open surjection.

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