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

Abstract

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 Latrémolière 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.