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Labeled Trees and Localized Automorphisms of the Cuntz Algebras

Authors
  • Conti, Roberto
  • Szymanski, Wojciech
Type
Published Article
Publication Date
Jun 24, 2010
Submission Date
May 29, 2008
Source
arXiv
License
Yellow
External links

Abstract

We initiate a detailed and systematic study of automorphisms of the Cuntz algebras $\O_n$ which preserve both the diagonal and the core $UHF$-subalgebra. A general criterion of invertibility of endomorphisms yielding such automorphisms is given. Combinatorial investigations of endomorphisms related to permutation matrices are presented. Key objects entering this analysis are labeled rooted trees equipped with additional data. Our analysis provides insight into the structure of ${\rm Aut}(\O_n)$ and leads to numerous new examples. In particular, we completely classify all such automorphisms of ${\mathcal O}_2$ for the permutation unitaries in $\otimes^4 M_2$. We show that the subgroup of ${\rm Out}(\O_2)$ generated by these automorphisms contains a copy of the infinite dihedral group ${\mathbb Z} \rtimes {\mathbb Z}_2$.

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