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

TL;DR

This paper studies the fixed-point algebra of the Cuntz algebra under cyclic automorphisms, proving its isomorphism to the original algebra and extending previous results to more cases.

Contribution

It demonstrates that the fixed-point algebra under cyclic permutation automorphisms is isomorphic to the original Cuntz algebra, generalizing prior work.

Findings

Fixed-point algebra of _n is *-isomorphic to _n

Extended the isomorphism result to _{2n} under exchange automorphism

Provided new structural insights into automorphisms of Cuntz algebras

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 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.