$G$-${\mathrm{C}^*}$-algebra. Then $\alpha$ has continuous Rokhlin property if and only if the dual action $\hat \alpha$ of $\hat{G}$ on $G \ltimes A$ is asymptotically representable.

2.  Let $(A,\alpha)$ be a ${\hat{G}}$-${\mathrm{C}^*}$-algebra. Then $\alpha$ is asymptotically representable if and only if the dual action $\hat \alpha$ of $G$ on ${\hat{G}}\ltimes A$ has continuous Rokhlin property.

For (1), first assume $\alpha$ has continuous Rokhlin property with the averaging map $\psi_t$. Set $$u_t := ({\mathrm{id}}\otimes \psi_t)(W \otimes 1_A) \in M(C^*_\lambda(G) \otimes A).$$ Since $\psi_t$ is $G$-equivariant and asymptotically central in $A$, we obtain $\hat{\alpha} (a) - u_t^* (1 \otimes a) u_t \to 0$.

Conversely assume $\hat \alpha$ is asymptotically representable with the implementing asymptotic representation $u_t \in M(C^*_\lambda(G) \otimes A)$. Then $u_t$ determines a unitary representation $u \in M(C^*_\lambda(G) \otimes \mathfrak{A}A)$, hence gives rise to a unital $*$-homomorphism $$\psi: C(G) \to \mathfrak{A}A.$$ Uniform continuity, $G$-equivariance and asymptotic centrality follow from the relation $\alpha(a) - u_t^*(1 \otimes a) u_t \to 0$.

The assertions in (2) follow from (1) and the Baaj-Skandalis duality.

The model action
----------------

Next, we construct a continuous Rokhlin action of a Hodgkin Lie group $G$ on a Kirchberg algebra which is ${\mathrm{KK}}^G$-equivalent to $C(G)$, more precisely, an asymptotically representable action 