of [@MR894590]).

In fact, ${\mathfrak{J}}_G$-injectivity is more relevant to asymptotic innerness of the dual action rather than asymptotic representability, which we need for continuous Rokhlin property (see Remark \[rmk:inner\]). For example, the crossed product of the infinite tensor product on a ${\mathrm{C}^*}$-algebra by the Bernoulli shift of ${\mathbb Z}$ is a ${\mathfrak{J}}_\mathbb{T}$-injective $\mathbb{T}$-${\mathrm{C}^*}$-algebra without continuous Rokhlin property.

\[lem:avbdl\] Let $A$ be a $G$-${\mathrm{C}^*}$-algebra with Rokhlin property. Then, there is an isomorphism $G \ltimes A \cong A^\alpha \otimes {\mathbb{K}}$. In particular, if $A$ and $B$ are $G$-${\mathrm{C}^*}$-algebras with Rokhlin property which are ${\mathrm{KK}}^G$-equivalent, then $A^\alpha$ and $B^\beta$ are ${\mathrm{KK}}$-equivalent.

It suffices to show that $A \otimes {\mathbb{K}}(L^2(G))\cong A \otimes {\mathbb{K}}$ because $(A \otimes {\mathbb{K}})^G=A^\alpha \otimes {\mathbb{K}}$ and $(A \otimes {\mathbb{K}}(L^2(G)))^G=G \ltimes A$. To this end, we show that for any two unitary representations ${\mathscr{H}}_0$ and ${\mathscr{H}}_1$ with the same (finite) dimension, ${\mathscr{H}}_0 \otimes A \cong {\mathscr{H}}_1 \otimes A$ as $G$-equivariant Hilbert $A$-modules.

Let $u_0$ be an equivariant isomorphism ${\mathscr{H}}_0 \otimes C(G)\to {\mathscr{H}}_1 \otimes C(G)$ and set $u:={\begin{pmatrix} 0 & u_0^* \\ u_0 & 0 \end{pmatrix}} \in {\mathbb{B}}({\mathscr{H}}) \otimes C(G)$ where ${\mathscr{H}}:={\mathscr{H}}_0 \oplus {\mathscr{H}}_1$. Since $A$ has Rokhlin property, there is a $G$-equivariant completely positive map $\varphi ':= \varphi \otimes {\mathrm{id}}_{{\mathbb{B}}({\mathscr{H}})} :C(G)\otimes 