such that $$(\pi_S(f) \xi_i, \xi_i) \to \int_G f(g) dg$$ for any $f \in C(G)$. Hence there exists $k \in {\mathbb Z}_{>0}$ and a unit vector $\xi \in \ell^2(S^{\otimes k})$ such that $$\| \langle ({\mathrm{id}}_{{\mathbb{B}}({\mathscr{H}}_\pi)} \otimes \pi_S^{\otimes k})(U_\pi)(1_{{\mathbb{B}}({\mathscr{H}}_\pi)} \otimes \xi),  1_{{\mathbb{B}}({\mathscr{H}}_\pi)} \otimes \xi \rangle \| < \varepsilon$$ for any $\pi \in F$, where $U_\pi \in {\mathbb{B}}({\mathscr{H}}_\pi)\otimes C(G)$ is the unitary representation of $G$ corresponding to $\pi$. Now we take the projection $p \in \mathbb{M}_n^{\otimes k}$ onto $\mathbb C \xi$. Using $$({\mathrm{id}}_{{\mathbb{B}}({\mathscr{H}}_\pi )} \otimes \pi_S^{\otimes k})(U_\pi) = (\pi \otimes {\mathrm{id}}_{\mathbb{M}_n^{\otimes k}})(U_S^{\otimes k}),$$ we get $p$ satisfies the desired property as follows: $$\begin{aligned}
& \| (\pi \otimes {\mathrm{id}})((1 \otimes p)(U_S^{\otimes k})^* (1 \otimes p)U_S^{\otimes k}) \|_{{\mathbb{B}}({\mathscr{H}}_\pi) \otimes \mathbb{M}_n^{\otimes k}} \\
= & \| \langle (\pi \otimes {\mathrm{id}})(U_S^{\otimes k})(1 \otimes \xi), 1 \otimes \xi \rangle ^* (1 \otimes p) (\pi \otimes {\mathrm{id}})(U_S^{\otimes k})  \|_{{\mathbb{B}}({\mathscr{H}}_\pi) \otimes \mathbb{M}_n^{\otimes k}} <  \varepsilon.\end{aligned}$$

Consider a unital embedding $$\mathbb{M}_n \oplus \mathbb{M}_{n+1} \to {\mathcal{O}}_\infty$$ and denote the canonical image of $U_S \oplus 1$ by $V$. This is a unitary representation of ${\hat{G}}$ on ${\mathcal{O}}_\infty$. Let $\displaystyle A := \bigotimes \nolimits_{k \in {\mathbb Z}_+} {\mathcal{O}}_\infty \simeq {\mathcal{O}}_\infty$ and consider the following action $\alpha$ of ${\hat{G}}$ on $A$ given by $$\alpha(x) = 