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

Published online by Cambridge University Press:  07 February 2011

ROBERTO CONTI*
Affiliation:
Dipartimento di Scienze, Università di Chieti-Pescara ‘G. D’Annunzio’, Viale Pindaro 42, I-65127 Pescara, Italy (email: [email protected])
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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.

Type
Research Article
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Archbold, R. J., ‘On the “flip-flop” automorphism of C *(S 1,S 2)’, Q. J. Math. 30 (1979), 129132.Google Scholar
[2]Conti, R. and Pinzari, C., ‘Remarks on endomorphisms of Cuntz algebras’, J. Funct. Anal. 142 (1996), 369405.Google Scholar
[3]Conti, R., Rørdam, M. and Szymański, W., ‘Endomorphisms of Ø n which preserve the canonical UHF-subalgebra’, J. Funct. Anal. 259 (2010), 602617.Google Scholar
[4]Conti, R. and Szymański, W., ‘Labeled trees and localized automorphisms of the Cuntz algebras’, Trans. Amer. Math. Soc., arXiv:0805.4654, to appear.Google Scholar
[5]Cuntz, J., ‘Simple C *-algebras generated by isometries’, Comm. Math. Phys. 57 (1977), 173185.Google Scholar
[6]Cuntz, J., ‘Automorphisms of certain simple C *-algebras’, in: Quantum Fields—Algebras, Processes (ed. Streit, L.) (Springer, Vienna–New York, 1980).Google Scholar
[7]Davidson, K. R., C *-Algebras by Example, Fields Institute Monographs, 6 (American Mathematical Society, Providence, RI, 1996).Google Scholar
[8]Evans, D. E. and Kawahigashi, Y., Quantum Symmetries on Operator Algebras, Oxford Mathematical Monographs (Oxford University Press, Oxford, 1998).Google Scholar
[9]Glimm, J. G., ‘On a certain class of operator algebras’, Trans. Amer. Math. Soc. 95 (1960), 318340.Google Scholar
[10]Power, S. C., ‘Homology for operator algebras, III. Partial isometry homotopy and triangular algebras’, New York J. Math. 4 (1998), 3556.Google Scholar