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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Labeled trees and localized automorphisms of the Cuntz algebras


Authors: Roberto Conti and Wojciech Szymański
Journal: Trans. Amer. Math. Soc. 363 (2011), 5847-5870
MSC (2010): Primary 46L40, 46L05, 37B10
Published electronically: June 2, 2011
MathSciNet review: 2817412
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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 $ {\rm 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 $ {\rm 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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Additional Information

Roberto Conti
Affiliation: Department of Mathematics, School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia
Address at time of publication: Department of Mathematics, University of Rome 2 Tor Vergata, Via della Ricerca Scientifica, 00133 Rome, Italy
Email: conti@mat.uniroma2.it

Wojciech Szymański
Affiliation: Department of Mathematics, School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia
Address at time of publication: Department of Mathematics and Computer Science, The University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark
Email: szymanski@imada.sdu.dk

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05234-7
PII: S 0002-9947(2011)05234-7
Keywords: Cuntz algebra, endomorphism, automorphism, Cartan subalgebra, core UHF-subalgebra, normalizer, permutation, tree
Received by editor(s): September 8, 2008
Received by editor(s) in revised form: October 15, 2009
Published electronically: June 2, 2011
Article copyright: © Copyright 2011 American Mathematical Society