A note on the Cuntz algebra automorphisms

TL;DR

This paper characterizes a specific class of involutions in the automorphisms of Cuntz algebras, confirming a conjecture and expanding the known family of automorphisms with new degrees of freedom.

Contribution

It proves a conjecture about stable involutions, introducing a new family of automorphisms of Cuntz algebras with six degrees of freedom.

Findings

Confirmed Conjecture 12.2 of Brenti and Conti

Identified a new family of automorphisms with 6 degrees of freedom

Characterized stable involutions of [n]^2

Abstract

Permutative automorphisms of the Cuntz algebras $\mathcal{O}_n$ are in bijection with the stable permutations of $[n]^k$. They are also the elements of the restricted Weyl group of $Aut(\mathcal{O}_n)$. In this note, we characterize a class of stable involutions of $[n]^2$. More precisely, we prove Conjecture 12.2 of Brenti and Conti [Adv. Math. 381 (2021), p. 60], and thus providing a new family (with $6$ degrees of freedom) of automorphisms of the Cuntz algebras $\mathcal{O}_n$ for any $n>1$.