see this, remind that a completely positive contractive map of order zero from $A$ to $B$ is of the form $\varphi (ta)$ where $\varphi$ is a unital $\ast$-homomorphism from ${\mathcal{C}}A:= C_0((0,1], A)^+$ to $B$ (Corollary 4.1 of [@MR2545617]). We remark that $\varphi (ta)$ is $G$-equivariant if and only if so does $\varphi$ and the join $X \ast Y$ of two compact spaces satisfies ${\mathcal{C}}(X \ast Y) \cong {\mathcal{C}}X \times {\mathcal{C}}Y$ where ${\mathcal{C}}X:= X \times [0,1]/{X \times \{ 0 \}}$.

\[prp:RokcRok\] For a $G$-${\mathrm{C}^*}$-algebra, we have $$\dim _{\mathrm{Rok}}^c (A,\alpha) \leq {\dim_{\mathrm{cRok}}}^c (A, \alpha ) \leq 2 \dim _{\mathrm{Rok}}^c (A,\alpha) +1.$$ In particular, we have ${\dim_{\mathrm{cRok}}}^c(A,\alpha ) \leq 1$ if $A$ has Rokhlin property.

We write $A_{\alpha, \infty}:=\ell ^\infty_\alpha ({\mathbb N}, A) /c_0({\mathbb N}, A) \cap A'$ as in [@mathOA14081946] and let $\varphi^i: C(G) \to A_{\alpha ,\infty}$ be completely positive maps of order zero such that $\varphi ^0(1)+\cdots +\varphi ^d(1)=1$. Let $F_n$ be an increasing sequence of finite subsets of $C(G)$ such that $\overline{\bigcup F_n}=C(G)$ and choose $\varphi _n$ inductively such that ${\left\| [\varphi ^i_n(f), \varphi ^i_m(f)] \right\|} \leq 2^{-n-m}$ for any $n,m \geq N$, $f \in F_N$ and $i=0,\dots, d$.

Then, we obtain $(2d+2)$ $G$-equivariant completely positive maps from $C(G)$ to 