The Weyl group of the Cuntz algebra

Abstract

The Weyl group of the Cuntz algebra $\mathcal{O}_n$ is investigated. This is (isomorphic to) the group of polynomial automorphisms $\lambda_u$ of $\mathcal{O}_n$, namely those induced by unitaries u that can be written as finite sums of words in the canonical generating isometries $S_i$ and their adjoints. A necessary and sufficient algorithmic combinatorial condition is found for deciding when a polynomial endomorphism $\lambda_u$ restricts to an automorphism of the canonical diagonal MASA. Some steps towards a general criterion for invertibility of $\lambda_u$ on the whole of $\mathcal{O}_n$ are also taken. A condition for verifying invertibility of a certain subclass of polynomial endomorphisms is given. First examples of polynomial automorphisms of $\mathcal{O}_n$ not inner related to permutative ones are exhibited, for every $n\geq 2$. In particular, the image of the Weyl group in the outer automorphism group of $\mathcal{O}_n$ is strictly larger than the image of the reduced Weyl group analyzed in previous papers. Results about the action of the Weyl group on the spectrum of the diagonal are also included.