Endomorphisms of the Cuntz Algebras

TL;DR

This paper explores the automorphisms and endomorphisms of Cuntz algebras, providing combinatorial descriptions, algebraic characterizations, and connections to classical dynamical systems, including new insights into the structure of these algebras.

Contribution

It offers a combinatorial framework for permutative automorphisms, characterizes the restricted Weyl group, and links it to classical dynamical systems, answering longstanding questions.

Findings

Permutative automorphisms described via labeled rooted trees

Restricted Weyl group related to classical dynamical systems on the Cantor set

Identification of automorphism groups with full two-sided n-shift for prime n

Abstract

This mainly expository article is devoted to recent advances in the study of dynamical aspects of the Cuntz algebras O_n, with n finite, via their automorphisms and, more generally, endomorphisms. A combinatorial description of permutative automorphisms of O_n in terms of labeled, rooted trees is presented. This in turn gives rise to an algebraic characterization of the restricted Weyl group of O_n. It is shown how this group is related to certain classical dynamical systems on the Cantor set. An identification of the image in Out(O_n) of the restricted Weyl group with the group of automorphisms of the full two-sided n-shift is given, for prime n, providing an answer to a question raised by Cuntz in 1980. Furthermore, we discuss proper endomorphisms of O_n which preserve either the canonical UHF-subalgebra or the diagonal MASA, and present methods for constructing exotic examples of such endomorphisms.