Fell convergence of the characters of the real line

Every irreducible strongly continuous unitary representation of the abelian group $\mathbb{R}$ is one-dimensional. Indeed, for fixed $g$ the operator $\pi (g)$ commutes with every $\pi (h)$, hence is a bounded self-intertwiner of $\pi$; [F1] makes it a scalar $\chi (g)I$. Every linear subspace is then invariant, so irreducibility forces $\dim H=1$; the resulting map $\chi :\mathbb{R}\to \mathbb{T}$ is a continuous unitary character.

By [F2] every continuous unitary character of $\mathbb{R}$ is ${\chi }_t$ for a unique $t\in \mathbb{R}$, and each ${\chi }_t$ is a continuous unitary character of $\mathbb{R}$.

Hence $\widehat{\mathbb{R}}=\{{\chi }_t:t\in \mathbb{R}\}$, with $t\mapsto {\chi }_t$ bijective: step 1.1 exhibits every irreducible class as a character, step 1.2 identifies the characters, and distinct $t$ give distinct characters (evaluate at a suitable $x$).

Fell convergence is compact-uniform convergence of the parameters. If $t_i\to t$, then for a compact $Q\subseteq \mathbb{R}$ and $R:={\sup }_{x\in Q}\mathrm{\mid }x\mathrm{\mid }$ one has ${\sup }_{x\in Q}\mathrm{\mid }{\chi }_{t_i}(x)-{\chi }_t(x)\mathrm{\mid }\le R\mathrm{\mid }{t}_i-t\mathrm{\mid }\to 0$; hence for every finite family $c_j{\chi }_t$ of tests, every compact $Q$ and every $ϵ>0$, the test $c_j{\chi }_t$ is within $ϵ$ on $Q$ of the coefficient $c_j{\chi }_{t_i}$ once $i$ is large, so ${\chi }_{t_i}\in W({\chi }_t;\cdot ,Q,ϵ)$ eventually and ${\chi }_{t_i}\to {\chi }_t$ in the Fell topology. Conversely, suppose ${\chi }_{t_i}\to {\chi }_t$, let $0<ϵ<1$ and fix $\delta >0$; put $Q:=[-\pi \mathrm{/}\delta ,\pi \mathrm{/}\delta ]$. By [F3] the neighbourhood determined by the coefficient ${\chi }_t$, the set $Q$ and $ϵ$ is met eventually: there are $c_i^{\mathrm{\prime }}\ge 0$ with ${\sup }_Q\mathrm{\mid }{\chi }_t-c_i^{\mathrm{\prime }}{\chi }_{t_i}\mathrm{\mid }<ϵ$. Evaluating at $x=0$ gives $\mathrm{\mid }1-c_i^{\mathrm{\prime }}\mathrm{\mid }<ϵ$, hence $c_i^{\mathrm{\prime }}>1-ϵ$. If $\mathrm{\mid }{t}_i-t\mathrm{\mid }\ge \delta$, then $x_i:=\pi \mathrm{/}\mathrm{\mid }{t}_i-t\mathrm{\mid }$ lies in $Q$ and ${\chi }_{t_i}(x_i)=-{\chi }_t(x_i)$, so $\mathrm{\mid }{\chi }_t(x_i)-c_i^{\mathrm{\prime }}{\chi }_{t_i}(x_i)\mathrm{\mid }=1+c_i^{\mathrm{\prime }}>2-ϵ>1>ϵ$, a contradiction. Hence eventually $\mathrm{\mid }{t}_i-t\mathrm{\mid }<\delta$, and since $\delta >0$ was arbitrary, $t_i\to t$.

Weak containment of characters: if ${\chi }_s\prec {\chi }_t$, then applying the defining approximation to the coefficient ${\chi }_s$ (the diagonal coefficient at a unit vector), the compact set $Q=[-\pi \mathrm{/}\delta ,\pi \mathrm{/}\delta ]$ and some $0<ϵ<1$, and using [F3], we find $c^{\mathrm{\prime }}\ge 0$ with ${\sup }_Q\mathrm{\mid }{\chi }_s-c^{\mathrm{\prime }}{\chi }_t\mathrm{\mid }<ϵ$; evaluating at $0$ gives $c^{\mathrm{\prime }}>1-ϵ$, and if $\mathrm{\mid }s-t\mathrm{\mid }\ge \delta$, the point $x=\pi \mathrm{/}\mathrm{\mid }s-t\mathrm{\mid }\in Q$ gives $\mathrm{\mid }{\chi }_s(x)-c^{\mathrm{\prime }}{\chi }_t(x)\mathrm{\mid }=1+c^{\mathrm{\prime }}>1>ϵ$, a contradiction. Hence $\mathrm{\mid }s-t\mathrm{\mid }<\delta$ for every $\delta >0$, so $s=t$; the converse is immediate by taking the identical coefficient. This agrees with The Fell closure of a single representation is its weak containment closure, since by step 3.1 the point ${\chi }_s$ lies in the Fell closure of $\{{\chi }_t\}$ exactly when $s=t$.

The Axiom of Choice is inherited from Schur's lemma and the Fell topology suppliers; the character and parameter computations use no further choice (The Axiom of Choice).