Automorphisms of the Cuntz algebras

TL;DR

This paper surveys recent advances in understanding automorphisms of Cuntz algebras, focusing on the Weyl group structure, endomorphisms preserving the diagonal, and combinatorial methods for permutative endomorphisms.

Contribution

It provides a comprehensive overview of the structure of automorphisms of Cuntz algebras, emphasizing the Weyl group and combinatorial approaches to permutative endomorphisms.

Findings

Characterization of automorphisms via the Weyl group

Analysis of endomorphisms preserving the diagonal MASA

Development of combinatorial methods for permutative endomorphisms

Abstract

We survey recent results on endomorphisms and especially on automorphisms of the Cuntz algebras O_n, with a special emphasis on the structure of the Weyl group. We discuss endomorphisms globally preserving the diagonal MASA and their corresponding actions. In particular, we investigate those endomorphisms of O_n which restrict to automorphisms of the diagonal. We review a combinatorial approach to the study of permutative endomorphisms. All the presented material is put in context with current research topics.