Endomorphisms of the Cuntz algebras

Research output: Contribution to journalConference articleResearchpeer-review

Abstract

This mainly expository article is devoted to recent advances in the study of dynamical aspects of the Cuntz algebras O_n, n<infty, 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.
Original languageEnglish
JournalBanach Center Publications
Volume96
Pages (from-to)81-97
ISSN0137-6934
DOIs
Publication statusPublished - 2012

Fingerprint

Cuntz Algebra
Endomorphisms
Automorphisms
Weyl Group
Rooted Trees
Cantor set
Subalgebra
Dynamical system

Keywords

  • Cunz algebra
  • endomorphism
  • automorphism

Cite this

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title = "Endomorphisms of the Cuntz algebras",
abstract = "This mainly expository article is devoted to recent advances in the study of dynamical aspects of the Cuntz algebras O_n, n<infty, 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.",
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author = "Roberto Conti and Hong, {Jeong Hee} and Wojciech Szymanski",
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}

Endomorphisms of the Cuntz algebras. / Conti, Roberto; Hong, Jeong Hee; Szymanski, Wojciech.

In: Banach Center Publications, Vol. 96, 2012, p. 81-97.

Research output: Contribution to journalConference articleResearchpeer-review

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AU - Hong, Jeong Hee

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AB - This mainly expository article is devoted to recent advances in the study of dynamical aspects of the Cuntz algebras O_n, n<infty, 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.

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