:= \overline{\mathrm{span}} (C(G) \otimes 1) \alpha(A) \subset {\mathbb{L}}(L^2(G) \otimes A).$$ We often omit $\alpha$ when no confusion arise. We also identify $A$ and $C(G)$ as subalgebras of ${\hat{G}}\ltimes A$. It is equipped with the dual $G$-action uniquely determined by ${\hat \alpha}_g(a)= a$ for $a \in A$ and ${\hat \alpha}_g(f)(x)= f(g^{-1} x) \in C(G)$ for $f \in C(G)$. The Baaj-Skandalis duality [@MR1235438] asserts that for any ${\hat{G}}$-${\mathrm{C}^*}$-algebra $A$, $G {\, }_{{\hat \alpha}}\! \ltimes ({\hat{G}}{\, }_{\alpha}\! \ltimes  A)$ is isomorphic to ${\mathbb{K}}(L^2(G)) \otimes A$ with the ${\hat{G}}$-action $$\tilde \alpha(a) = W_{12}^* ({\mathrm{id}}\otimes \alpha)(a)_{213} W_{12}$$ as $G$-${\mathrm{C}^*}$-algebras.

We use these terminologies in order to generalize a characterization of continuous Rokhlin property in terms of the dual action of ${\hat{G}}$ on $G \ltimes A$ as in Lemma 3.8 of [@MR2053753] for general compact groups.

We say that an action $\alpha$ of ${\hat{G}}$ on $A$ is *asymptotically representable* if there is a strictly continuous path of (asymptotic) unitaries $u_t \in M(C^*_\lambda(G) \otimes M(A)^\alpha)$ which satisfies $$\begin{aligned}
({\hat \Delta}\otimes {\mathrm{id}})(u_t) - (u_t)_{13} (u_t)_{23} \to &0,\\
\alpha(a) - u_t^* (1 \otimes a) u_t \to &0\end{aligned}$$ for any $a \in A$ strictly as $t \to \infty$.

\[lem:asymprep\] Let $G$ be a compact group.

1.  Let $(A,\alpha)$ be a 