Primitive ideal space

For a $C^*$ -algebra $A$ , its primitive ideal space is the set of primitive ideals

\operatorname{Prim}(A)=\{P\subsetneq A:P\text{ is primitive}\}

equipped with the hull–kernel topology. For $S\subseteq A$ , define

\operatorname{hull}(S)=\{P\in\operatorname{Prim}(A):S\subseteq P\}.

The closed subsets are exactly the sets $\operatorname{hull}(I)$ , where $I$ ranges over closed two-sided ideals of $A$ . Equivalently, for $Y\subseteq\operatorname{Prim}(A)$ ,

\overline Y=\operatorname{hull}\left(\bigcap_{P\in Y}P\right).

This topology is canonically determined by the ideal structure of $A$ .

Ideals and open subsets

The hull–kernel topology turns the ideal structure of $A$ into topology. For a closed two-sided ideal $I$ , $\operatorname{hull}(I)$ is naturally homeomorphic to $\operatorname{Prim}(A/I)$ , while

\operatorname{Prim}(A)\setminus\operatorname{hull}(I)

is naturally homeomorphic to $\operatorname{Prim}(I)$ . In fact, closed two-sided ideals of $A$ correspond order-preservingly to open subsets of $\operatorname{Prim}(A)$ through this construction.

Relation to irreducible representations

Let $\widehat A$ denote the unitary-equivalence classes of nonzero irreducible representations of $A$ . The kernel map

\widehat A\longrightarrow\operatorname{Prim}(A),\qquad [\pi]\longmapsto\ker\pi,

is always surjective, but it need not be injective. For type I $C^*$ -algebras, an irreducible representation is determined up to unitary equivalence by its kernel, so the distinction disappears at the level of sets.

Commutative case and separation

If $A=C_0(X)$ for a locally compact Hausdorff space $X$ , evaluation at each $x\in X$ has kernel

P_x=\{f:f(x)=0\},

and $x\mapsto P_x$ is a homeomorphism $X\cong\operatorname{Prim}(A)$ . This explains why primitive ideals serve as noncommutative points.

In general, $\operatorname{Prim}(A)$ is a $T_0$ space but need not be Hausdorff or even $T_1$ . These failures are meaningful: specialization relations record inclusions among primitive ideals. The phrase $C^*$ -spectrum is sometimes used for $\operatorname{Prim}(A)$ , but it must not be confused with the spectrum $\sigma_A(a)$ of a single element.

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. DOI record. Relevant: Chapter 4 on primitive spectra and Chapter 6 on type I representation theory.