Approximate representability of finite abelian group actions on the Razak-Jacelon algebra

Let $A$ be a simple separable nuclear monotracial C${}^{\ast }$-algebra, and let $\alpha$ be an outer action of a finite abelian group $\mathrm{\Gamma }$ on $A$. In this paper, we show that $\alpha \otimes {\mathrm{i}\mathrm{d}}_{\mathcal{W}}$ on $A\otimes \mathcal{W}$ is approximately representable if and only if the characteristic invariant of $\tilde{\alpha }$ is trivial, where $\mathcal{W}$ is the Razak-Jacelon algebra and $\tilde{\alpha }$ is the induced action on the injective II${}_1$ factor ${\pi }_{{\tau }_A}(A{)}^{{}^{\mathrm{\prime }\mathrm{\prime }}}$. As an application of this result, we classify such actions up to conjugacy and cocycle conjugacy. In particular, we show the following: Let $A$ and $B$ be simple separable nuclear monotracial C${}^{\ast }$-algebras, and let $\alpha$ and $\beta$ be outer actions of a finite abelian group $\mathrm{\Gamma }$ on $A$ and $B$, respectively. Assume that the characteristic invariants of $\tilde{\alpha }$ and $\tilde{\beta }$ are trivial. Then $\alpha \otimes {\mathrm{i}\mathrm{d}}_{\mathcal{W}}$ and $\beta \otimes {\mathrm{i}\mathrm{d}}_{\mathcal{W}}$ are conjugate (resp. cocycle conjugate) if and only if $\tilde{\alpha }$ on ${\pi }_{{\tau }_A}(A{)}^{{}^{\mathrm{\prime }\mathrm{\prime }}}$ and $\tilde{\beta }$ on ${\pi }_{{\tau }_B}(B{)}^{{}^{\mathrm{\prime }\mathrm{\prime }}}$ are conjugate (resp. cocycle conjugate). We also construct the model actions.