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AUTOMORPHISMS OF THE UHF ALGEBRA THAT DO NOT EXTEND TO THE CUNTZ ALGEBRA

Part of: Topological dynamics Selfadjoint operator algebras

Published online by Cambridge University Press:  07 February 2011

ROBERTO CONTI
ROBERTO CONTI*
Affiliation:
Dipartimento di Scienze, Università di Chieti-Pescara ‘G. D’Annunzio’, Viale Pindaro 42, I-65127 Pescara, Italy (email: conti@sci.unich.it)
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Abstract

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The automorphisms of the canonical core UHF subalgebra ℱn of the Cuntz algebra 𝒪n do not necessarily extend to automorphisms of 𝒪n. Simple examples are discussed within the family of infinite tensor products of (inner) automorphisms of the matrix algebras Mn. In that case, necessary and sufficient conditions for the extension property are presented. Also addressed is the problem of extending to 𝒪n the automorphisms of the diagonal 𝒟n, which is a regular maximal abelian subalgebra with Cantor spectrum. In particular, it is shown that there exist product-type automorphisms of 𝒟n that do not extend to (possibly proper) endomorphisms of 𝒪n.

Keywords

Cuntz algebraendomorphismautomorphismcore UHF subalgebraregular maximal abelian subalgebra

MSC classification

Secondary: 46L40: Automorphisms 37B10: Symbolic dynamics
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 89 , Issue 3 , December 2010 , pp. 309 - 315
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

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