Fell neighbourhoods of an irreducible representation are saturated under weak equivalence

Let $T\subseteq \widehat{G}$ be a set of classes such that every standard Fell neighbourhood of $\pi$ meets $T$. Then $\pi \prec {\widehat{⨁}}_{\sigma \in T}\sigma$. Indeed, let $\varphi$ be a function of positive type associated to $\pi$, $Q$ compact and $ϵ>0$; the standard neighbourhood $W(\pi ;\varphi ,Q,ϵ)$ contains a class in $T$, which by definition of $W$ supplies a finite sum of functions of positive type associated to that member of $T$, and such a finite sum is a finite sum of functions of positive type associated to the direct sum over $T$ (each is computed from finitely many vectors supported on finitely many summands); this is precisely the defining approximation for $\pi \prec {\widehat{⨁}}_{\sigma \in T}\sigma$.

Every $\tilde{W}(\pi ;{\varphi }_1,\dots ,{\varphi }_n,Q,ϵ)$ containing $\pi$ contains a standard Fell neighbourhood of $\pi$. Suppose, to the contrary, that no standard neighbourhood of $\pi$ is contained in $\tilde{W}:=\tilde{W}(\pi ;{\varphi }_1,\dots ,{\varphi }_n,Q,ϵ)$; then every standard neighbourhood of $\pi$ meets $T:=\widehat{G}\setminus \tilde{W}$, so $\pi \prec {\widehat{⨁}}_{\sigma \in T}\sigma$ by step 1.1. By [F5] choose representatives of the classes in $T$. Each tested function is a single diagonal coefficient, so write ${\varphi }_i(g)=⟨\pi (g){\xi }_i,{\xi }_i⟩$. Applying [F4] to the finite list ${\xi }_1,\dots ,{\xi }_n$ on $Q$ with radius $ϵ$ gives a single class $\sigma \in T$ and vectors ${\eta }_i\in H_{\sigma }$ such that each ${\varphi }_i$ is within $ϵ$ on $Q$ of the single coefficient $⟨\sigma (\cdot ){\eta }_i,{\eta }_i⟩$. This says $\sigma \in \tilde{W}$, contradicting $\sigma \in \widehat{G}\setminus \tilde{W}$. Hence $\tilde{W}$ contains a standard neighbourhood of $\pi$, and with [F1] the sets $\tilde{W}$ containing $\pi$ form a neighbourhood basis.

Every Fell-open set is saturated under weak equivalence. Let $U$ be Fell-open and $\pi \in U$; by the definition of the Fell topology and step 2.1 there are ${\varphi }_1,\dots ,{\varphi }_n,Q,ϵ$ with $\pi \in \tilde{W}(\pi ;{\varphi }_1,\dots ,{\varphi }_n,Q,ϵ)\subseteq U$. Let $\rho \in \widehat{G}$ with $\rho \sim \pi$, so in particular $\pi \prec \rho$; since $\pi$ is irreducible, write ${\varphi }_i=⟨\pi (\cdot ){\xi }_i,{\xi }_i⟩$. If ${\xi }_i=0$, use the zero vector of $H_{\rho }$. Otherwise apply [F3] to ${\varphi }_i\mathrm{/}\mathrm{\parallel }{\xi }_i{\mathrm{\parallel }}^2$ with precision $ϵ\mathrm{/}\mathrm{\parallel }{\xi }_i{\mathrm{\parallel }}^2$ and rescale its unit-vector witness by $\mathrm{\parallel }{\xi }_i\mathrm{\parallel }$. This supplies a single diagonal coefficient of $\rho$ within $ϵ$ of each ${\varphi }_i$ on $Q$. Hence $\rho \in \tilde{W}\subseteq U$. Thus $U$ contains the weak equivalence class of each of its points, and by symmetry the same holds for $\rho$ in place of $\pi$.

The Axiom of Choice licenses the choice of representatives of the classes in $T$ through the unitary dual and is inherited from the family-selection and approximation lemmas; the contradiction argument and the saturation computation add no further choice (The Axiom of Choice).