Equivariant Kirchberg-Phillips type absorption for the Razak-Jacelon algebra

Let $A$ and $B$ be simple separable nuclear monotracial C${}^{\ast }$-algebras, and let $\alpha$ and $\beta$ be strongly outer actions of a countable discrete amenable group $\mathrm{\Gamma }$ on $A$ and $B$, respectively. In this paper, we show that $\alpha \otimes {\mathrm{i}\mathrm{d}}_{\mathcal{W}}$ on $A\otimes \mathcal{W}$ and $\beta \otimes {\mathrm{i}\mathrm{d}}_{\mathcal{W}}$ on $B\otimes \mathcal{W}$ are cocycle conjugate where $\mathcal{W}$ is the Razak-Jacelon algebra. Also, we characterize such actions by using the fixed point subalgebras of Kirchberg’s central sequence C${}^{\ast }$-algebras.