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

We investigate the structure of the fixed-point algebra of On under the action of the cyclic permutation of the generating isometries. We prove that it is ⁎-isomorphic with On, thus generalizing a result of Choi and Latrémolière on O2. As an application of the technique employed, we also describe the fixed-point algebra of O2n under the exchange automorphism.