Rokhlin actions of finite groups on UHF-absorbing $\mathrm {C}^*$-algebras

Abstract:

This paper serves as a source of examples of Rokhlin actions or locally representable actions of finite groups on $\mathrm {C}^*$-algebras satisfying a certain UHF-absorption condition. We show that given any finite group $G$ and a separable, unital $\mathrm {C}^*$-algebra $A$ that absorbs $M_{|G|^\infty }$ tensorially, one can lift any group homomorphism $G\to \mathrm {Aut}(A)/_{{\approx _{\mathrm {u}}}}$ to an honest Rokhlin action $\gamma$ of $G$ on $A$. Unitality may be dropped in favour of stable rank one or being stable. If $A$ belongs to a certain class of $\mathrm {C}^*$-algebras that is classifiable by a suitable invariant (e.g. $K$-theory), then in fact every $G$-action on the invariant lifts to a Rokhlin action of $G$ on $A$. For the crossed product $\mathrm {C}^*$-algebra ${A\rtimes _\gamma G}$ of a Rokhlin action on a UHF-absorbing $\mathrm {C}^*$-algebra, an inductive limit decomposition is obtained in terms of $A$ and $\gamma$. If $G$ is assumed to be abelian, then the dual action $\hat {\gamma }$ is locally representable in a very strong sense. We then show how some well-known constructions of finite group actions with certain predescribed properties can be recovered and extended by the main results of this paper when paired with known classification theorems. Among these is Blackadar’s famous construction of symmetries on the CAR algebra whose fixed point algebras have non-trivial $K_1$-groups. Lastly, we use the results of this paper to reduce the UCT problem for separable, nuclear $\mathrm {C}^*$-algebras to a question about certain finite group actions on $\mathcal {O}_2$.