This mainly expository article is devoted to recent advances in the study of dynamical aspects of the Cuntz algebras 𝓞ₙ, n < ∞, via their automorphisms and, more generally, endomorphisms. A combinatorial description of permutative automorphisms of 𝓞ₙ in terms of labelled, rooted trees is presented. This in turn gives rise to an algebraic characterization of the restricted Weyl group of 𝓞ₙ. It is shown how this group is related to certain classical dynamical systems on the Cantor set. An identification of the image in Out(𝓞ₙ) 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 𝓞ₙ which preserve either the canonical UHF-subalgebra or the diagonal MASA, and present methods for constructing exotic examples of such endomorphisms.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc96-0-5,
author = {Roberto Conti and Jeong Hee Hong and Wojciech Szyma\'nski},
title = {Endomorphisms of the Cuntz algebras},
journal = {Banach Center Publications},
volume = {95},
year = {2011},
pages = {81-97},
zbl = {1259.46047},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc96-0-5}
}
Roberto Conti; Jeong Hee Hong; Wojciech Szymański. Endomorphisms of the Cuntz algebras. Banach Center Publications, Tome 95 (2011) pp. 81-97. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc96-0-5/