Labeled trees and localized automorphisms of the Cuntz algebras

Conti, Roberto; Szymański, Wojciech

doi:10.1090/S0002-9947-2011-05234-7

Labeled trees and localized automorphisms of the Cuntz algebras
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by Roberto Conti and Wojciech Szymański PDF

Trans. Amer. Math. Soc. 363 (2011), 5847-5870 Request permission

Abstract:

We initiate a detailed and systematic study of automorphisms of the Cuntz algebras $\mathcal {O}_n$ 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 $\textrm {Aut}(\mathcal {O}_n)$ 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 $\textrm {Out}(\mathcal {O}_2)$ generated by these automorphisms contains a copy of the infinite dihedral group ${\mathbb Z} \rtimes {\mathbb Z}_2$.

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