The induced kernel map on weak equivalence classes is a homeomorphism

$\kappa$ is well defined, surjective, and its fibres are the weak equivalence classes. If $[\pi ]=[\rho ]$ then $\pi$ and $\rho$ are unitarily equivalent, hence have equal kernels, so $\kappa$ is well defined. A primitive ideal is by definition the kernel of an irreducible nondegenerate star-representation of $C^{\ast }(G)$; by [F2] it is the kernel of the extension of an irreducible unitary representation $\pi$ of $G$, so it equals $\kappa ([\pi ])$, proving surjectivity. Finally $\kappa ([\pi ])=\kappa ([\rho ])$ means $\ker \pi =\ker \rho$, which by [F3] is equivalent to $\pi \sim \rho$.

If $\pi \in \bar{S}$ for $S\subseteq \widehat{G}$, then $\pi \prec {\widehat{⨁}}_{\sigma \in S}\sigma$. Every standard Fell neighbourhood of $\pi$ meets $S$ by definition of the closure [F5]; given a function of positive type $\varphi$ associated to $\pi$, a compact $Q$ and $ϵ>0$, the neighbourhood $W(\pi ;\varphi ,Q,ϵ)$ contains some $\sigma \in S$, so $\varphi$ is within $ϵ$ on $Q$ of a finite sum of functions of positive type associated to $\sigma$, hence of a finite sum of functions of positive type associated to the direct sum; this is the defining approximation for weak containment [F8]. (For $S=\mathrm{\varnothing }$ the hypothesis $\pi \in \bar{S}$ is false, so there is nothing to prove.)

If $\pi \prec {\widehat{⨁}}_{\sigma \in S}\sigma$ for $S\subseteq \widehat{G}$, then $\pi \in \bar{S}$. Apply [F6] to the family of representatives of the classes in $S$ and to each standard test: for finitely many tested single coefficients ${\varphi }_i(g)=⟨\pi (g){\xi }_i,{\xi }_i⟩$, compact $Q$ and $ϵ>0$, the simultaneous selection yields a single $s\in S$ and vectors ${\eta }_i\in H_{{\sigma }_s}$ with ${\sup }_Q\mathrm{\mid }{\varphi }_i(g)-⟨{\sigma }_s(g){\eta }_i,{\eta }_i⟩\mathrm{\mid }<ϵ$. Each approximating coefficient is an allowed one-term sum; hence ${\sigma }_s$ lies in the tested neighbourhood and every Fell neighbourhood of $\pi$ meets $S$. (When $S=\mathrm{\varnothing }$ the weak containment $\pi \prec 0$ is impossible, since it would force the kernel $C^{\ast }(G)$ of the zero representation into $\ker \pi$ and $\pi =0$, contrary to irreducibility.)

Closure identity: for every $S\subseteq \widehat{G}$, $\bar{S}={\kappa }^{-1}\left({\overline{\kappa (S)}}^{\mathrm{J}\mathrm{a}\mathrm{c}}\right)$. By steps 1.2 and 1.3, $\bar{S}=\{\pi :\pi \prec {\widehat{⨁}}_{\sigma \in S}\sigma \}$; by [F3] and the kernel computation of [F2], $\pi \prec {\widehat{⨁}}_{\sigma \in S}\sigma$ is equivalent to ${\bigcap }_{\sigma \in S}\kappa (\sigma )\subseteq \kappa (\pi )$; and by [F4] the set of primitive ideals containing ${\bigcap }_{\sigma \in S}\kappa (\sigma )$ is exactly the Jacobson closure of $\kappa (S)$ (for $S=\mathrm{\varnothing }$ both sides are empty, since ${\bigcap }_{\mathrm{\varnothing }}=C^{\ast }(G)$ and no primitive ideal contains $C^{\ast }(G)$, by [F4], while $\bar{\mathrm{\varnothing }}=\mathrm{\varnothing }$).

$\kappa$ is continuous. Let $D\subseteq \mathrm{Prim}(C^{\ast }(G))$ be Jacobson closed and put $S:={\kappa }^{-1}(D)$. Then $\kappa (S)=D$ by the surjectivity of step 1.1, so by step 2.1 $\bar{S}={\kappa }^{-1}\left({\bar{D}}^{\mathrm{J}\mathrm{a}\mathrm{c}}\right)={\kappa }^{-1}(D)=S$; hence ${\kappa }^{-1}(D)$ is Fell closed and $\kappa$ is continuous.

$\kappa$ is closed. Let $S\subseteq \widehat{G}$ be Fell closed, so $\bar{S}=S$; by step 2.1, $S={\kappa }^{-1}\left({\overline{\kappa (S)}}^{\mathrm{J}\mathrm{a}\mathrm{c}}\right)$, and applying the surjective $\kappa$ to both sides gives $\kappa (S)=\kappa \left({\kappa }^{-1}\right({\overline{\kappa (S)}}^{\mathrm{J}\mathrm{a}\mathrm{c}}\left)\right)={\overline{\kappa (S)}}^{\mathrm{J}\mathrm{a}\mathrm{c}}$ by step 1.1; hence $\kappa (S)$ is Jacobson closed.

The induced bijection is a homeomorphism. The fibres of $\kappa$ are the weak equivalence classes by step 1.1, so $\kappa$ induces a bijection $\bar{\kappa }$ from the set of classes, equipped with the quotient Fell topology along $q:\widehat{G}\to \widehat{G}\mathrm{/}\sim$, onto $\mathrm{Prim}(C^{\ast }(G))$. Since $\kappa =\bar{\kappa }\circ q$ is continuous by step 3.1, the universal property of the quotient topology [F7] makes $\bar{\kappa }$ continuous. If $E\subseteq \widehat{G}\mathrm{/}\sim$ is closed, then $q^{-1}(E)$ is Fell closed by definition of the quotient topology and $\bar{\kappa }(E)=\kappa (q^{-1}(E))$ (as $q$ is surjective) is Jacobson closed by step 3.2; hence $\bar{\kappa }$ is a continuous closed bijection, that is, a homeomorphism.

The Axiom of Choice is inherited from the representation correspondence, the weak-containment suppliers and the family-selection lemma; the closure identity, the topology argument and the quotient identification add no further choice (The Axiom of Choice).