such that for any $z \in X$ $$\begin{aligned}
&\| xzx \| \geq (1 - \varepsilon) \|z\|, \\
&\| (1 \otimes xz)\alpha_\pi(x) \| < \varepsilon \| z \|.\end{aligned}$$

We may assume $X \subset ({\mathcal{O}}_\infty)^{\otimes k}$. Pick $p \in (\mathbb{M}_n)^{\otimes l}$ as in Lemma \[Lemma\]. Consider the inclusion $$(\mathbb{M}_n)^{\otimes l} \subset (\mathbb{M}_n \oplus \mathbb{M}_{n+1})^{\otimes l} \subset ({\mathcal{O}}_\infty)^{\otimes l}$$ and let us denote the image of $p$ by $q$. Then $q$ is a nonzero projection such that $$\| (1_{{\mathbb{B}}({\mathscr{H}}_\pi)} \otimes q) \alpha_\pi(q) \| < \varepsilon$$ for any $\pi \in F$. Now, $$x := 1_{{\mathcal{O}}_\infty^{\otimes k}} \otimes q \in ({\mathcal{O}}_\infty)^{\otimes (k+l)}$$ satisfies the property. In fact, since $x$ commutes with any $z \in X$, the first condition is obvious and the second one follows from $$(1 \otimes xz)\alpha_\pi(x) = (1 \otimes z) (V^{\otimes k})^* ((1 \otimes q)\alpha_\pi(q))_{1(k+2)(k+3)\ldots(k+l+1)} V^{\otimes k}.$$

The ${\mathrm{C}^*}$-algebra ${\hat{G}}{\, }_{\alpha}\! \ltimes A$ is purely infinite.

For any positive $a \in G {\, }_{\alpha}\! \ltimes A $, we approximate $a$ by positive $$c = \sum_{\pi,i,j} c^\pi_{ij} u^\pi_{ij} \in {\hat{G}}{\, }_{\alpha, \mathrm{alg}}\! \ltimes A$$ such that $\| a - c \| < \varepsilon$, where $u_{ij}^\pi$ are matrix coefficients of $\pi$ with respect to a fixed basis and ${\hat{G}}{\, }_{\alpha, \mathrm{alg}}\! \ltimes A$ is 