the $*$-subalgebra of ${\hat{G}}\ltimes A$ generated by the polynomial algebra $\mathrm{Pol}(G)$ of $G$ and $A$.

We may assume $\| c^1 \| = 1$. Let $F \subset {\rm Irr}(G)$ be the support of $c$, $n$ the number of nonzero $c^\pi_{ij}$’s. Applying the previous lemma for $X = \{c^\pi_{ij} \}$ to get $x \in A_+$ such that $$\begin{aligned}
&\| x c^1 x \| \geq 1 - \varepsilon , \\
&\| (1 \otimes x c^\pi_{ij}) \alpha_\pi(x) \| < \varepsilon / n.\end{aligned}$$ Hence we get $\|x c x - x c^1 x \| < 1 - 2 \varepsilon$. The rest of the argument is completely the same as Lemma 10 in [@MR1424962].

Classification
--------------

An important feature of Kirchberg algebras is the Kirchberg-Phillips classification: two Kirchberg algebras are isomorphic if and only if there is a ${\mathrm{KK}}$-equivalence between them preserving unit classes (Theorem 4.2.4 of [@MR1745197]). It is generalized for equivariant setting when the actions have Rokhlin property.

\[lem:intw\] Let $A$ and $B$ be $G$-${\mathrm{C}^*}$-algebras with $G$-actions $\alpha$ and $\beta$. Assume $B$ has Rokhlin property and we have a $*$-homomorphism $\varphi: A \to B$ such that there exists a sequence of unitaries $(u_n)_{n \in \mathbb{N}} \in B \otimes C(G)$ such that $$\operatorname{Ad}(u_n(g)) \circ \varphi(x)\to 