Symmetry in the Cuntz Algebra on two generators

TL;DR

This paper studies the automorphism of the Cuntz algebra 2 that swaps its generators, revealing that its fixed point subalgebra is isomorphic to 2 and exploring the connection with the crossed product.

Contribution

It demonstrates that the fixed point algebra under the generator-swapping automorphism is isomorphic to 2, providing new insights into the algebra's symmetry structure.

Findings

Fixed point subalgebra is isomorphic to 2

Detailed relationship between crossed product and fixed point algebra

Automorphism exchanging generators has a well-understood fixed point structure

Abstract

We investigate the structure of the automorphism of $\mathcal{O}_{2}$ which exchanges the two canonical isometries. Our main observation is that the fixed point C*-subalgebra for this action is isomorphic to $\mathcal{O}_{2}$ and we detail the relationship between the crossed-product and fixed point subalgebra.