is the *central path algebra* of $(A,\alpha)$.

Let $G$ be a compact group and let $X$ be a compact $G$-space.

1.  We say that $(A,\alpha)$ has *continuous Rokhlin property* if there is an $G$-equivariant unital $\ast$-homomorphism $\varphi : C(G) \to \mathfrak{C}_\alpha A$.

2.  We say that $(A,\alpha)$ has *continuous $X$-Rokhlin property* if there is an $G$-equivariant unital $\ast$-homomorphism $\varphi :C(X) \to \mathfrak{C}_\alpha A$.

3.  We say that $(A,\alpha)$ has *continuous Rokhlin dimension* less than $d$ *with commuting tower* and write as ${\dim_{\mathrm{cRok}}}^c (A,\alpha)\leq d$ if there are completely positive contractive order zero maps $\varphi ^0, \dots , \varphi ^d: C(G) \to \mathfrak{C}_\alpha A$ with commuting ranges such that $\varphi ^0(1)+\dots +\varphi ^d(1)=1$.

Here we say that this $\varphi$ is a continuous Rokhlin map for $(A,\alpha)$.

These are variations of Rokhlin property (Definition 3.1 of [@MR2053753]) and the Rokhlin dimension (Definition 2.3 of [@mathOA12091618]), which are defined by using ${\mathbb Z}_{>0}$ instead of $[0,\infty)$.

A $G$-${\mathrm{C}^*}$-algebra $A$ has continuous Rokhlin dimension less than $d-1$ with commuting tower if and only if $A$ has continuous $E_dG$-Rokhlin property where $E_dG:=G \ast \cdots \ast G$ is the $d$-th step of the Milnor construction (cf. Lemma 1.7 of [@MR3398712] and Lemma 4.4 of [@mathOA14071277]). To 