Fixed-point permanence under actions by finite quantum groups

Abstract Given an action by a finite quantum group $\mathbb {G}$ on a von Neumann algebra M , we prove that a number of familiar $W^*$ properties are equivalent for M and the fixed-point algebra $M^{\mathbb {G}}$ (i.e., hold or not simultaneously for the two algebras); these include being hyperfinite, atomic, diffuse, and of type I , $II$ , or $III$ . Moreover, in all cases, the canonical central projections of M and $M^{\mathbb {G}}$ cutting out the summand with the respective property coincide. The result generalizes its classical- $\mathbb {G}$ analog due to Jones–Takesaki.