More localized automorphisms of the Cuntz algebras

TL;DR

This paper classifies localized automorphisms of Cuntz algebras $O_n$ for $n=3,4$ using combinatorial and computational methods, extending previous analyses for $O_2$ and exploring diagonal automorphisms.

Contribution

It provides a complete classification of certain localized automorphisms of $O_3$ and $O_4$, employing novel combinatorial and computational techniques.

Findings

Classified localized automorphisms for $O_3$ and $O_4$

Determined automorphisms of diagonals induced by permutation matrices

Extended analysis framework from $O_2$ to higher $n$

Abstract

We completely determine the localized automorphisms of the Cuntz algebras $O_n$ corresponding to permutation matrices in $M_n \otimes M_n$ for $n=3$ and $n=4$. This result is obtained through a combination of general combinatorial techniques and large scale computer calculations. Our analysis proceeds according to the general scheme proposed in a previous paper, where we analyzed in detail the case of $O_2$ using labeled rooted trees. We also discuss those proper endomorphisms of these Cuntz algebras which restrict to automorphisms of their respective diagonals. In the case of $O_3$ we compute the number of automorphisms of the diagonal induced by permutation matrices in $M_3 \otimes M_3 \otimes M_3$.