On the cyclic automorphism of the Cuntz algebra and its fixed-point algebra

We investigate the structure of the fixed-point algebra of $\mathcal{O}_n$ under the action of the cyclic permutation of the generating isometries. We prove that it is $*$-isomorphic with $\mathcal{O}_n$, thus generalizing a result of Choi and Latr\'emoli\`ere on $\mathcal{O}_2$. As an application of the technique employed, we also describe the fixed-point algebra of $\mathcal{O}_{2n}$ under the exchange automorphism.