Endomorphisms of the Cuntz algebras
Tom 96 / 2011
Streszczenie
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.
