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Endomorphisms of the Cuntz algebras. (English) Zbl 1259.46047

Bożejko, Marek (ed.) et al., Noncommutative harmonic analysis with applications to probability. III: Papers presented at the 13th workshop, Bȩdlewo, Poland, July 11–17, 2010. Warszawa: Polish Academy of Sciences, Institute of Mathematics (ISBN 978-83-86806-15-7/pbk). Banach Center Publications 96, 81-97 (2012).
Summary: 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 J. Cuntz [in: Quantum fields - algebras, processes, Proc. Symp., Bielefeld 1978, 187–196 (1980; Zbl 0475.46046)]. 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.
For the entire collection see [Zbl 1248.46002].

MSC:

46L05 General theory of \(C^*\)-algebras
46L40 Automorphisms of selfadjoint operator algebras
37A55 Dynamical systems and the theory of \(C^*\)-algebras

Citations:

Zbl 0475.46046
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