$\mathfrak{C}_\alpha A$ of order zero given by $$\tilde{\varphi} _{2n+j+t}^{i,j}(f)=(1-|t|)\varphi ^i_n(f) \ \  \text{ for $t \in [-1,1]$}$$ for $i=0,\dots ,d$ and $j=0,1$. By definition, the images in $\mathfrak{C}_\alpha A$ commute and $\sum \tilde{\varphi}^{i,j}_t(1)=1$.

Continuous Rokhlin property is actually strictly stronger than Rokhlin property. See Subsection \[section:5.3\] for more detail.

The following examples are pointed out by Eusebio Gardella. Let $G$ be a finite group. The UHF algebra $\mathbb{M}_{|G|^\infty}$ with the $G$-action $\alpha _g :=\bigotimes ^\infty \operatorname{Ad}\lambda _g$ has continuous Rokhlin property. Actually, we obtain an asymptotically central path of mutually orthogonal projections $\{ p_t^g \}_{g \in G}$ satisfying $\alpha _h(p_t^g)=p^{hg}_t$ given by $$p_t^g:=1 \otimes \cdots \otimes 1 \otimes u_{t-n} (p_g \otimes 1) u_{t-n}^* \otimes 1 \otimes  \cdots$$ for $t \in [n,n+1]$. Here $p_g \in \mathbb{M}_{|G|}$ is the projection corresponding to $g \in G$ and $u_t$ is a homotopy of $G$-invariant unitaries in $\mathbb{M}_{|G|}^{\otimes 2}$ such that $u_0=1$ and $u_1$ is the flip on ${\mathbb C}^{|G|} \otimes {\mathbb C}^{|G|}$.

Since ${\mathcal{O}}_n \otimes \mathbb{M}_{n^\infty} \cong {\mathcal{O}}_n$, we obtain an example of continuous Rokhlin actions on the Cuntz algebras $\mathcal{O}_{|G|}$. On the other hand, since any automorphism on ${\mathcal{O}}_{|G|}$ is approximately inner (Theorem 3.6 of [@MR1225963]), Theorem 3.5 of [@MR2053753] implies that every 