Mathematics – Operator Algebras
Scientific paper
2008-05-30
Trans. Amer. Math. Soc. 363, 5847-5870 (2011)
Mathematics
Operator Algebras
35 pages, slight changes, to appear on Trans. Amer. Math. Soc
Scientific paper
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 ${\mathbb Z} \rtimes {\mathbb Z}_2$.
Conti Roberto
Szymanski Wojciech
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