The primitive ideal space of a group C star algebra

Assume the Axiom of Choice. Let $G$ be an LCH group. A closed two-sided ideal $I⊴C^{\ast }(G)$ is primitive if it is the kernel of an irreducible nondegenerate star-representation of $C^{\ast }(G)$ (C star algebra, Nondegenerate representations of the full group C star algebra are unitary representations, The full (maximal) group C star algebra). The primitive ideal space $\mathrm{Prim}(C^{\ast }(G))$ is the set of primitive ideals, equipped with the Jacobson topology, whose closed sets are the sets $h(J):=\{I\in \mathrm{Prim}(C^{\ast }(G)):I\supseteq J\}$ for closed two-sided ideals $J⊴C^{\ast }(G)$.

Under the correspondence between unitary representations of $G$ and nondegenerate star-representations of $C^{\ast }(G)$, an irreducible unitary representation $\pi$ determines the primitive ideal $C^{\ast }\ker \pi :={\ker }_{C^{\ast }(G)}\pi$, and the assignment $\kappa :\widehat{G}\to \mathrm{Prim}(C^{\ast }(G)),\kappa ([\pi ]):=C^{\ast }\ker \pi ,$ is well defined on unitary equivalence classes (The unitary dual of a locally compact group).

• Kernels of equivalent representations agree. If $U:H_{\pi }\to H_{\rho }$ is a unitary intertwiner, then $\pi (f)=U^{-1}\rho (f)U$ for every $f\in L^1(G)$ and, by continuity of the extensions, for every element of $C^{\ast }(G)$; hence $\ker \pi =\ker \rho$ and $\kappa$ is well defined on classes.
• The Jacobson closed sets satisfy the topology axioms. Finite intersections of hulls are hulls of the closed ideals generated by the union; arbitrary intersections are hulls of the ideal generated by the union; $h(0)$ is the whole space and $h(C^{\ast }(G))=\mathrm{\varnothing }$. Finite unions use that primitive ideals are prime: if $J_1{J}_2\subseteq P=\ker \pi$ for an irreducible $\pi$, then $\pi (J_1)H_{\pi }$ and $\pi (J_2)H_{\pi }$ are ideals images; if both were nonzero they would be dense invariant subspaces (irreducibility), and $\pi (J_1)\pi (J_2)H_{\pi }$ would be dense and zero at once; hence $J_1\subseteq P$ or $J_2\subseteq P$. Consequently $h(J_1)\cup h(J_2)=h(J_1\cap J_2)$, and the displayed family of closed sets is a topology. No assertion that primitive ideals are maximal is used.
• Comparison with the Fell topology. The set $\mathrm{Prim}(C^{\ast }(G))$ with the Jacobson topology carries the quotient topology induced by $\kappa$ from the Fell topology on $\widehat{G}$ when the comparison is established; that identification is proved by the kernel-map theorem later on this page and is not assumed here (The Fell topology on the unitary dual).
• Choice. The Axiom of Choice is inherited from the representation correspondence; taking kernels and forming hulls uses no further choice (The Axiom of Choice).