Hull–kernel topology - Imported Knowlpedia Content

Definition

Let $A$ be a $C^*$ -algebra and $\operatorname{Prim}(A)$ its set of primitive ideals. For $S\subseteq A$ and $Y\subseteq\operatorname{Prim}(A)$ , define

\operatorname{hull}(S)=\{P\in\operatorname{Prim}(A):S\subseteq P\}, \qquad \ker(Y)=\bigcap_{P\in Y}P.

The hull–kernel topology, or Jacobson topology, is the topology whose closed sets are the hulls of closed two-sided ideals. Equivalently,

\overline{Y}=\operatorname{hull}(\ker(Y)).

With the convention $\ker(\varnothing)=A$ , this formula also gives the closure of the empty set. The two operations reverse inclusions, and their composite on subsets of $\operatorname{Prim}(A)$ is precisely the topological closure operator.

Ideal–open-set correspondence

For a closed two-sided ideal $I\triangleleft A$ , write

U_I=\operatorname{Prim}(A)\setminus\operatorname{hull}(I).

The assignment $I\mapsto U_I$ is an order-preserving bijection from closed two-sided ideals of $A$ to open subsets of $\operatorname{Prim}(A)$ . Under this correspondence, $\operatorname{hull}(I)$ is naturally the primitive ideal space of the quotient $A/I$ , while $U_I$ is naturally the primitive ideal space of $I$ Dixmier, §3.1.

Separation and specialization

The space is always $T_0$ , but it need not be $T_1$ or Hausdorff. Indeed,

\overline{\{P\}}=\{Q\in\operatorname{Prim}(A):P\subseteq Q\}.

Thus the specialization relation remembers inclusions among primitive ideals. When $A=C_0(X)$ for a locally compact Hausdorff space $X$ , the map

x\longmapsto\{f\in C_0(X):f(x)=0\}

identifies $X$ homeomorphically with the primitive ideal space, so the hull–kernel topology recovers the original topology.

Why “kernel”

Primitive ideals are kernels of irreducible representations. Intersecting the members of $Y$ extracts the algebraic information common to those representations, and taking the hull returns every primitive ideal that contains it. The topology is therefore controlled by the ideal lattice rather than by a metric on representations.

References

Jacques Dixmier, $C^*$ -Algebras, North-Holland Mathematical Library 15, North-Holland, 1977. Publisher record. Relevant: §3.1 on primitive ideals and the hull–kernel topology.
Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. Publisher DOI record. Relevant: §4.1 on primitive spectra and ideal correspondences.