We say that a countable discrete group action $\alpha$ on a C$^*$-algebra $A$ is \textit{ $\mathcal{ W}$-absorbing} if there exist a C$^*$-algebra $B$ and an action $\beta$ on $B$ such that $\alpha$ is cocycle conjugate to $\beta\otimes \mathrm{ id}_{ \mathcal{ W}}$ on $B\otimes \mathcal{ W}$ where $\mathcal{ W}$ is the Razak-Jacelon algebra. In this paper, we completely classify outer $\mathcal{ W}$-absorbing actions of finite groups on $\mathcal{ W}$ up to conjugacy and cocycle conjugacy.
Categories: math.OA