is also available for the study of Rokhlin property because the Rokhlin dimension with commuting tower is finite if and only if so is the continuous Rokhlin dimension with commuting tower (Proposition \[prp:RokcRok\]).

Although continuous Rokhlin property is strictly stronger than Rokhlin property, many of known examples of Rokhlin actions have continuous Rokhlin property. Actually, a necessary and sufficient condition for a Rokhlin action on a Kirchberg algebra to have continuous Rokhlin property is given in terms of equivariant ${\mathrm{KK}}$-theory (Proposition \[prp:injcRok\]).

Our first main theorem (Theorem \[thm:KKHod\]) asserts that any $G$-${\mathrm{C}^*}$-algebra with continuous Rokhlin property is ${\mathrm{KK}}^G$-equivalent to $A^\alpha \otimes C(G)$ when $G$ is a compact Lie group with Hodgkin condition (connected and $\pi _1(G)$ is torsion-free). To see this, we use the strong Baum-Connes isomorphism for an arbitrary coefficient of the dual quantum group ${\hat{G}}$ of a Hodgkin Lie group $G$ [@MR2339371], which is rephrased in terms of the crossed product functor since ${\hat{G}}$ is torsion-free in the sense of [@MR2563811].

Next, we apply it for classification of Kirchberg $G$-algebras with continuous Rokhlin property. In [@MR2053753], Izumi gives a necessary and sufficient condition for two actions with Rokhlin property to be conjugate by using an intertwining argument. By 