A positive rank-one Haar average is a nonzero compact intertwiner

The map $R_{\xi }x=⟨x,\xi ⟩\xi$ is linear and bounded with $\mathrm{\parallel }{R}_{\xi }x\mathrm{\parallel }\le \mathrm{\parallel }x\mathrm{\parallel }\mathrm{\parallel }\xi {\mathrm{\parallel }}^2$ by Cauchy--Schwarz, and $\mathrm{\parallel }{R}_{\xi }\xi \mathrm{\parallel }=\mathrm{\parallel }\xi {\mathrm{\parallel }}^3$, so $\mathrm{\parallel }{R}_{\xi }\mathrm{\parallel }=\mathrm{\parallel }\xi {\mathrm{\parallel }}^2$; its range is contained in the span $\mathbb{C}\xi$ of the one-term list $\xi$, which admits the ordered basis of length one given by $\xi$ because $\xi \mathrm{\ne }0$, so $R_{\xi }$ is compact by [A4]. Moreover $⟨R_{\xi }x,y⟩=⟨x,\xi ⟩⟨\xi ,y⟩=⟨x,⟨y,\xi ⟩\xi ⟩=⟨x,R_{\xi }y⟩$, so $R_{\xi }$ is self-adjoint, $⟨R_{\xi }x,x⟩=\mathrm{\mid }⟨x,\xi ⟩{\mathrm{\mid }}^2\ge 0$, and $⟨R_{\xi }\xi ,\xi ⟩=\mathrm{\parallel }\xi {\mathrm{\parallel }}^4>0$, so $R_{\xi }\mathrm{\ne }0$.

For every $k\in K$ the operator $\phi (k)=\pi (k)R_{\xi }\pi (k{)}^{-1}$ is bounded; it is compact by [A4], because $\phi (k)x=⟨x,\pi (k)\xi ⟩\pi (k)\xi$ has range in the one-dimensional span of the nonzero vector $\pi (k)\xi$; it is self-adjoint and non-negative because $⟨\phi (k)x,y⟩=⟨R_{\xi }\pi (k{)}^{-1}x,\pi (k{)}^{-1}y⟩=⟨\pi (k{)}^{-1}x,R_{\xi }\pi (k{)}^{-1}y⟩=⟨x,\phi (k)y⟩$ and $⟨\phi (k)x,x⟩=⟨R_{\xi }\pi (k{)}^{-1}x,\pi (k{)}^{-1}x⟩\ge 0$ by step 1.1; and unitary conjugation preserves norms, so $\mathrm{\parallel }\phi (k)\mathrm{\parallel }=\mathrm{\parallel }{R}_{\xi }\mathrm{\parallel }=\mathrm{\parallel }\xi {\mathrm{\parallel }}^2$. The orbit map $k\mapsto \phi (k)$ is continuous in operator norm by [A5], since $R_{\xi }$ is bounded of finite rank.

The orbit $\phi :K\to \mathcal{B}(H)$ is continuous by step 2.1 into the Banach space $\mathcal{B}(H)$, so its image $\phi [K]$ is a compact subset of the metric space $\mathcal{B}(H)$ and hence totally bounded: for each $n\ge 1$ there is a finite $F_n\subseteq \phi [K]$ with $\phi [K]\subseteq {\bigcup }_{y\in F_n}B(y,1\mathrm{/}n)$; choosing these nets for all $n$, and enumerating each finite net, is licensed by Countable Choice and finite choice. List $F_n=\{y_0,\dots ,y_m\}$ and put $A_j:={\phi }^{-1}[B(y_j,1\mathrm{/}n)]\setminus {\bigcup }_{i<j}{A}_i$: each $A_j$ is Borel, being a difference of Borel sets, the $A_j$ are pairwise disjoint, and they cover $K$; hence $t_n:={\sum }_j{y}_j{\mathbf{1}}_{A_j}$ is a $\mathcal{B}(H)$-valued measurable simple function, and $\mathrm{\parallel }{t}_n(k)-\phi (k)\mathrm{\parallel }<1\mathrm{/}n$ for every $k\in K$, because $k$ lies in the first $A_j$ whose net point is within $1\mathrm{/}n$ of $\phi (k)$. Thus the $t_n$ converge to $\phi$ pointwise in norm, so $\phi$ is strongly measurable; and since $\mathrm{\parallel }\phi (k)\mathrm{\parallel }=\mathrm{\parallel }\xi {\mathrm{\parallel }}^2$ for every $k$ and $\mu (K)=1$, one has ${\int }_K\mathrm{\parallel }\phi \mathrm{\parallel }d\mu =\mathrm{\parallel }\xi {\mathrm{\parallel }}^2<\mathrm{\infty }$ and ${\int }_K\mathrm{\parallel }\phi -t_n\mathrm{\parallel }d\mu \le 1\mathrm{/}n\to 0$, so $\phi$ is Bochner integrable and $Q_{\xi }:={\int }_K\phi d\mu \in \mathcal{B}(H)$ is defined with $\mathrm{\parallel }{Q}_{\xi }\mathrm{\parallel }\le {\int }_K\mathrm{\parallel }\phi \mathrm{\parallel }d\mu =\mathrm{\parallel }\xi {\mathrm{\parallel }}^2$.

For all $v,w\in H$ the pairing formula holds: $⟨Q_{\xi }v,w⟩={\int }_K⟨\phi (k)v,w⟩d\mu (k)$, by applying the commuting theorem of [A6] to the bounded linear functional $S\mapsto ⟨Sv,w⟩$ on $\mathcal{B}(H)$ and the integrable function $\phi$.

The operator $Q_{\xi }$ is compact. By the norm inequality of [A6], $\mathrm{\parallel }{Q}_{\xi }-{\int }_K{t}_{n}d\mu \mathrm{\parallel }\le {\int }_K\mathrm{\parallel }\phi -t_n\mathrm{\parallel }d\mu \le 1\mathrm{/}n$, so $Q_{\xi }$ is the norm limit of the integrals ${\int }_K{t}_{n}d\mu$. Each of those is ${\sum }_j\mu (A_j)y_j$, a finite linear combination of net points $y_j\in \phi [K]$, and each such point equals $\phi (k_j)$ for some $k_j\in K$ and is therefore compact by step 2.1; so each ${\int }_K{t}_{n}d\mu$ is compact by [A8], and the norm limit $Q_{\xi }$ is compact by [A8] under Countable Choice.

The operator $Q_{\xi }$ is self-adjoint with $⟨Q_{\xi }x,x⟩\ge 0$ for every $x$: by step 4.1 and the pointwise properties of step 2.1, $⟨Q_{\xi }x,x⟩={\int }_K⟨\phi (k)x,x⟩d\mu (k)\ge 0$, and $⟨Q_{\xi }x,y⟩={\int }_K⟨\phi (k)x,y⟩d\mu (k)={\int }_K⟨x,\phi (k)y⟩d\mu (k)=\overline{{\int }_K⟨\phi (k)y,x⟩d\mu (k)}=\overline{⟨Q_{\xi }y,x⟩}=⟨x,Q_{\xi }y⟩$.

$Q_{\xi }\mathrm{\ne }0$: by steps 2.1 and 4.1, $⟨Q_{\xi }\xi ,\xi ⟩={\int }_K\mathrm{\mid }⟨\pi (k{)}^{-1}\xi ,\xi ⟩{\mathrm{\mid }}^{2}d\mu (k)$. The integrand is continuous, non-negative, and equals $\mathrm{\parallel }\xi {\mathrm{\parallel }}^4>0$ at $k=e$, so by continuity there is an open neighbourhood $U$ of $e$ on which it exceeds $\frac{1}{2}\mathrm{\parallel }\xi {\mathrm{\parallel }}^4$; then the integral is at least $\frac{1}{2}\mathrm{\parallel }\xi {\mathrm{\parallel }}^4\mu (U)>0$ by positivity of $\mu$ on nonempty open sets, since $U$ is nonempty. Hence $⟨Q_{\xi }\xi ,\xi ⟩>0$ and in particular $Q_{\xi }\mathrm{\ne }0$.

$Q_{\xi }$ commutes with $\pi (K)$: for $g\in K$ and $v,w\in H$, the function $k\mapsto ⟨\phi (k)\pi (g)v,w⟩$ is continuous and bounded on compact $K$ by step 2.1, hence integrable against the probability $\mu$. Using unitarity, the relation $\pi (g)\phi (k)=\phi (gk)\pi (g)$, and the translation invariance of the scalar integral, $⟨\pi (g)Q_{\xi }v,w⟩=⟨Q_{\xi }v,\pi (g{)}^{-1}w⟩={\int }_K⟨\phi (k)v,\pi (g{)}^{-1}w⟩d\mu (k)={\int }_K⟨\pi (g)\phi (k)v,w⟩d\mu (k)={\int }_K⟨\phi (gk)\pi (g)v,w⟩d\mu (k)={\int }_K⟨\phi (k)\pi (g)v,w⟩d\mu (k)=⟨Q_{\xi }\pi (g)v,w⟩$; since this holds for all $v,w$, the operators agree.

Collecting: $\phi$ is norm continuous and Bochner integrable and $Q_{\xi }={\int }_K\phi d\mu$ is a bounded self-adjoint operator with $⟨Q_{\xi }x,x⟩\ge 0$ for all $x$ (step 5.1), nonzero (step 5.2), compact (step 4.2), and commuting with every $\pi (g)$ (step 5.3). The Axiom of Choice entered only through the normalized Haar measure of [A1], the countable selections in the Bochner and net constructions of step 3.1, and the countable-choice compact-operator limit of [A8].