Endomorphisms of the Cuntz algebras

Tom 96 / 2011

Roberto Conti, Jeong Hee Hong, Wojciech Szymański Banach Center Publications 96 (2011), 81-97 MSC: 46L05, 46L40. DOI: 10.4064/bc96-0-5


This mainly expository article is devoted to recent advances in the study of dynamical aspects of the Cuntz algebras $\mathcal O_n$, $n < \infty$, via their automorphisms and, more generally, endomorphisms. A combinatorial description of permutative automorphisms of $\mathcal O_n$ in terms of labelled, rooted trees is presented. This in turn gives rise to an algebraic characterization of the restricted Weyl group of $\mathcal O_n$. It is shown how this group is related to certain classical dynamical systems on the Cantor set. An identification of the image in $\operatorname{Out}(\mathcal O_n)$ 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 $\mathcal O_n$ which preserve either the canonical UHF-subalgebra or the diagonal MASA, and present methods for constructing exotic examples of such endomorphisms.


  • Roberto ContiDipartimento di Scienze di Base e Applicate per l'Ingegneria
    Sezione di Matematica, Sapienza Università di Roma
    Via A. Scarpa 16, 00161 Roma, Italy
  • Jeong Hee HongDepartment of Data Information, Korea Maritime University
    Busan 606-791, South Korea
  • Wojciech SzymańskiDepartment of Mathematics and Computer Science, The University of Southern Denmark
    Campusvej 55, DK-5230 Odense M, Denmark

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