Irreducible weak containment in a family selects one coefficient

If $S=\mathrm{\varnothing }$ the direct sum is the zero representation and $\pi \prec 0$ is impossible, because the unit coefficient $⟨\pi (\cdot )\xi ,\xi ⟩$ takes the value $1$ at $e$ and cannot be approximated by $0$ on the compact set $\{e\}$; hence $S\mathrm{\ne }\mathrm{\varnothing }$ and $F:=\{{\omega }_{{\rho }_s,\eta }:s\in S,\eta \in H_{{\rho }_s},\mathrm{\parallel }\eta \mathrm{\parallel }=1\}$ is a nonempty subset of $K$ by [F3]. The functional $\omega :={\omega }_{\pi ,\xi }$ lies in $K$, has norm $1$ and is extreme in $K$ by [F4].

For every compact $Q\subseteq G$ and $ϵ>0$, a convex combination of $F$ approximates $\omega$ within $ϵ$ on $Q$. Fix $0<\delta <1\mathrm{/}4$ so that $4\delta \mathrm{/}(1-2\delta )<ϵ$, and apply [F1] on $Q\cup \{e\}$ with error $\delta$. This gives finitely many vectors ${\eta }_l$ in the Hilbert direct sum. Truncate each ${\eta }_l$ to finitely many summands so that the sum of the uniform coefficient errors is $<\delta$: this is possible by norm density of finitely supported vectors and the estimate $\mathrm{\mid }{c}_{{\eta }_l,{\eta }_l}-c_{v_l,v_l}\mathrm{\mid }\le (\mathrm{\parallel }{\eta }_l\mathrm{\parallel }+\mathrm{\parallel }{v}_l\mathrm{\parallel })\mathrm{\parallel }{\eta }_l-v_l\mathrm{\parallel }$. Write the resulting finite coefficient sum as $\psi (g)={\sum }_{l,s}⟨{\rho }_s(g)v_l^{(s)},v_l^{(s)}⟩$, retaining each pair $(l,s)$ separately. Then ${\sup }_{Q\cup \{e\}}\mathrm{\mid }\psi -\omega \mathrm{\mid }<2\delta$. Put $t={\sum }_{l,s}\mathrm{\parallel }{v}_l^{(s)}{\mathrm{\parallel }}^2=\psi (e)$, so $\mathrm{\mid }t-1\mathrm{\mid }<2\delta$ and $t>0$. Dividing each nonzero vector by its norm shows that $\varphi :=\psi \mathrm{/}t$ is the convex combination of normalized vector states with weights $\mathrm{\parallel }{v}_l^{(s)}{\mathrm{\parallel }}^2\mathrm{/}t$. Finally $\mathrm{\mid }\varphi -\omega \mathrm{\mid }\le (\mathrm{\mid }\psi -\omega \mathrm{\mid }+\mathrm{\mid }1-t\mathrm{\mid }\mathrm{\mid }\omega \mathrm{\mid })\mathrm{/}t<4\delta \mathrm{/}(1-2\delta )<ϵ$ on $Q$.

The functional $\omega$ lies in the weak-* closure $C$ of $\mathrm{co}(F)$ in $A^{\ast }$. Let $f_1,\dots ,f_N\in L^1(G)$ and $\eta >0$. By [F8] choose compactly supported continuous $g_i$ with $\mathrm{\parallel }{f}_i-g_i{\mathrm{\parallel }}_1<\eta \mathrm{/}12$, and put $Q:={\bigcup }_i\mathrm{supp}{g}_i$, a compact set with $2{\int }_{G\setminus Q}\mathrm{\mid }{f}_i\mathrm{\mid }<\eta \mathrm{/}3$ for every $i$. By step 2.1 choose $\varphi \in \mathrm{co}(F)$ with ${\sup }_Q\mathrm{\mid }\varphi -\omega \mathrm{\mid }<\eta \mathrm{/}(3+3{\sum }_i\mathrm{\parallel }{f}_i{\mathrm{\parallel }}_1)$; then, since $\mathrm{\mid }\varphi -\omega \mathrm{\mid }\le 2$ on $G$ and each $\varphi \in \mathrm{co}(F)$ lies in $K$ by convexity [F4], $\mid {\int }_G{f}_i(\varphi -\omega )\mid \le \mathrm{\parallel }{f}_i{\mathrm{\parallel }}_1{\sup }_Q\mathrm{\mid }\varphi -\omega \mathrm{\mid }+2{\int }_{G\setminus Q}\mathrm{\mid }{f}_i\mathrm{\mid }<\eta$ for every $i$. The image of $L^1(G)$ is dense in $A$ and $\mathrm{\parallel }\varphi \mathrm{\parallel },\mathrm{\parallel }\omega \mathrm{\parallel }\le 1$, so the same conclusion holds with $f_i$ replaced by arbitrary $a_i\in A$: every weak-* neighbourhood of $\omega$ meets $\mathrm{co}(F)$, that is, $\omega \in C$.

Since $C=\stackrel{}{\mathrm{co}}(F)\subseteq K$ is weak-* closed and convex and $K$ is compact by [F4], $C$ is compact. By step 1.1 the functional $\omega$ is extreme in $K$ and $\omega \in C$ by step 3.1, so $\omega$ is extreme in $C$; Milman's converse [F5] applied to the compact convex set $C=\stackrel{}{\mathrm{co}}(F)$ gives $\omega \in \bar{F}$. Hence there is a net $({\varphi }_{\alpha })\subseteq F$ converging to $\omega$ in the weak-* topology of $A^{\ast }$.

Part 1 of the statement holds. Each ${\varphi }_{\alpha }$ is ${\omega }_{{\rho }_{s(\alpha )},\eta (\alpha )}$ for some $s(\alpha )\in S$ and unit vector $\eta (\alpha )\in H_{{\rho }_{s(\alpha )}}$; its restriction to $L^1(G)$ is integration against the normalized coefficient ${\psi }_{\alpha }(g)=⟨{\rho }_{s(\alpha )}(g)\eta (\alpha ),\eta (\alpha )⟩\in P_1(G)$, and ${\int }_{G}f{\psi }_{\alpha }\to {\int }_{G}f{\omega }_{\pi ,\xi }(g)dg$ for every $f\in L^1(G)$ because ${\varphi }_{\alpha }\to \omega$ weak-* on $A$. By Raikov's theorem [F6] the net $({\psi }_{\alpha })$ converges to the coefficient $⟨\pi (\cdot )\xi ,\xi ⟩$ uniformly on compact subsets, so for the prescribed compact $Q$ and $ϵ$ some $\alpha$ has ${\sup }_Q\mathrm{\mid }{\psi }_{\alpha }(g)-⟨\pi (g)\xi ,\xi ⟩\mathrm{\mid }<ϵ$; with $s=s(\alpha )$ and $\eta =\eta (\alpha )$ this is the first assertion.

Part 2 of the statement holds. If the tested vector list is empty, choose any $s\in S$, which is nonempty by step 1.1, and the assertion is vacuous. Otherwise fix a unit vector $\xi \in H_{\pi }$, which exists since $\pi$ is irreducible and hence acts on a nonzero space. For each $i$ with ${\xi }_i\mathrm{\ne }0$, cyclicity [F9] (applied to the nonzero vector $\xi$) gives $c_{ij}\in \mathbb{C}$ and $x_{ij}\in G$ with $v_i:={\sum }_j{c}_{ij}\pi (x_{ij})\xi$ satisfying $\mathrm{\parallel }{\xi }_i-v_i\mathrm{\parallel }<\min (1,ϵ\mathrm{/}(4(1+\mathrm{\parallel }{\xi }_i\mathrm{\parallel })))$. Then $\mathrm{\parallel }{v}_i\mathrm{\parallel }\le \mathrm{\parallel }{\xi }_i\mathrm{\parallel }+1$, so $(\mathrm{\parallel }{\xi }_i\mathrm{\parallel }+\mathrm{\parallel }{v}_i\mathrm{\parallel })\mathrm{\parallel }{\xi }_i-v_i\mathrm{\parallel }<ϵ\mathrm{/}2$ by [F10]; for ${\xi }_i=0$ take $v_i=0$. Put $M_i=({\sum }_j\mathrm{\mid }{c}_{ij}\mathrm{\mid }{)}^2$ and $M=1+{\max }_i{M}_i$, and apply part 1 to the compact set ${\bigcup }_{i,j,k}{x}_{ik}^{-1}Q{x}_{ij}$ with radius $\delta =ϵ\mathrm{/}(2M)$: this gives $s\in S$ and a unit vector $\eta \in H_{{\rho }_s}$ with $\sup \mathrm{\mid }⟨\pi (h)\xi ,\xi ⟩-⟨{\rho }_s(h)\eta ,\eta ⟩\mathrm{\mid }<\delta$ over that union. Set $w_i:={\sum }_j{c}_{ij}{\rho }_s(x_{ij})\eta$. Expanding, $⟨\pi (g)v_i,v_i⟩={\sum }_{j,k}{c}_{ij}{\bar{c}}_{ik}⟨\pi (x_{ik}^{-1}g{x}_{ij})\xi ,\xi ⟩$ and $⟨{\rho }_s(g)w_i,w_i⟩={\sum }_{j,k}{c}_{ij}{\bar{c}}_{ik}⟨{\rho }_s(x_{ik}^{-1}g{x}_{ij})\eta ,\eta ⟩$, so on $Q$ the two coefficients differ by at most $\delta M_i<ϵ\mathrm{/}2$ by [F2] and [F10]; combined with the perturbation bound $\mathrm{\mid }⟨\pi (g){\xi }_i,{\xi }_i⟩-⟨\pi (g)v_i,v_i⟩\mathrm{\mid }\le (\mathrm{\parallel }{\xi }_i\mathrm{\parallel }+\mathrm{\parallel }{v}_i\mathrm{\parallel })\mathrm{\parallel }{\xi }_i-v_i\mathrm{\parallel }<ϵ\mathrm{/}2$ of [F10] this yields ${\sup }_Q\mathrm{\mid }⟨\pi (g){\xi }_i,{\xi }_i⟩-⟨{\rho }_s(g)w_i,w_i⟩\mathrm{\mid }<ϵ$ for every $i$.

Part 3 of the statement holds. Let $W(\pi ;{\varphi }_1,\dots ,{\varphi }_N,Q,ϵ)$ be a basic Fell neighbourhood of $[\pi ]$ [F7]; if $N=0$ the neighbourhood is the whole dual and meets the nonempty family. Otherwise write each tested function as ${\varphi }_i(g)=⟨\pi (g){\xi }_i,{\xi }_i⟩$. Applying step 6.1 to ${\xi }_1,\dots ,{\xi }_N$ with precision $ϵ$ gives one $s\in S$ and vectors ${\eta }_i\in H_{{\rho }_s}$ whose individual diagonal coefficients approximate the corresponding ${\varphi }_i$ on $Q$ within $ϵ$; each is an allowed one-term approximating sum. Hence $[{\rho }_s]\in W$; since every basic neighbourhood of $[\pi ]$ is met by $\{[{\rho }_s]:s\in S\}$, the class $[\pi ]$ lies in the closure of that set when all ${\rho }_s$ are irreducible (so that their classes lie in the dual).

The Axiom of Choice is inherited from the extreme-point supplier, the Hilbert direct sum, Raikov's theorem and the finitely many group elements selected in the cyclicity argument; the coefficient and convexity computations are choice-free (The Axiom of Choice).