The Weyl group of the Cuntz algebra

TL;DR

This paper explores the structure and automorphisms of the Weyl group of the Cuntz algebra O_n, providing algorithms for automorphism criteria, examples of non-inner automorphisms, and analyzing their action on the spectrum.

Contribution

It introduces a combinatorial algorithm for automorphism detection, exhibits new polynomial automorphisms, and compares Weyl group images in automorphism groups of O_n.

Findings

A necessary and sufficient condition for polynomial automorphisms to restrict to the diagonal.

Examples of polynomial automorphisms not inner related to permutative automorphisms.

The Weyl group's image in the outer automorphism group is larger than previously analyzed.

Abstract

The Weyl group of the Cuntz algebra O_n, with n finite, is investigated. This is (isomorphic to) the group of polynomial automorphisms of O_n, namely those induced by unitaries that can be written as finite sums of words in the canonical generating isometries and their adjoints. A necessary and sufficient algorithmic combinatorial condition is found for deciding when a polynomial endomorphism restricts to an automorphism of the canonical diagonal MASA. Some steps towards a general criterion for invertibility of such endomorphisms on the whole of 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 O_n not inner related to permutative ones are exhibited, for every n. In particular, the image of the Weyl group in the outer automorphism group of 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.