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Operator algebras with a reduction property

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  09 April 2009

James A. Gifford
James A. Gifford
Affiliation:
Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia, e-mail: giffordj@maths.anu.edu.au
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Abstract

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Given a representation θ: A → B(H) of a Banach algebra A on a Hilbert space H, H is said to have the reduction property as an A—module if every closed invariant subspace of H is complemented by a closed invariant subspace; A has the total reduction property if for every representation θ: A → B(H), H has the reduction property.

We show that a C*—algebra has the total reduction property if and only if all its representations are similar to *—representations. The question of whether all C*-algebras have this property is the famous ‘similarity problem’ of Kadison.

We conjecture that non-self-adjoint operator algebras with the total reduction property are always isomorphic to C*-algebras, and prove this result for operator algebras consisting of compact operators.

MSC classification

Secondary: 46L05: General theory of $C^*$-algebras 46L07: Operator spaces and completely bounded maps
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 80 , Issue 3 , June 2006 , pp. 297 - 315
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Arveson, W., An invitation to C*-algebra (Springer, New York, 1976).CrossRefGoogle Scholar
[2]Davidson, K. R., Nest algebras (Longman Scientific and Technical, Essex, 1988).Google Scholar
[3]Day, M. M., Normed linear spaces, 3rd edition (Springer, Berlin, 1973).CrossRefGoogle Scholar
[4]‘Amenable operators on Hilbert spaces’, J. Reine Angew. Math. 582 (2005), 201–228.Google Scholar
[5]Haagerup, U. ‘Solution of the similarity problem for cyclic representations of C*-algebras’, Ann. of Math. (2) 118 (1983), 215–240.CrossRefGoogle Scholar
[6]Johnson, B. E., Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 (Amer. Math. Soc., Providence, RI, 1972).CrossRefGoogle Scholar
[7]Kadison, R. V. ‘On the orthogonalization of operator representations’, Amer. J. Math. 77 (1955), 600–620.CrossRefGoogle Scholar
[8]lau, A. T. -M., Loy, R. J. and Willis, G. A. ‘Amenability of Banach and C*-algebras on locally compact groups’, Studia Math. 119 (1996), 161–178.Google Scholar
[9]Paulsen, V., Completely bounded maps and dilations (Longman Scientific and Technical, Essex, 1986).Google Scholar
[10]Pisier, G., Similarity problems and completely bounded maps, Lecture Notes in Math. 1 (Springer, 1996).CrossRefGoogle Scholar
[11]Radjavi, H. and Rosenthal, P., Invariant subspaces (Springer, Berlin, 1973).CrossRefGoogle Scholar
[12]Rosenoer, S. ‘Completely reducible algebras containing compact operators’, J. Operator Theory 29 (1993), 269–285.Google Scholar
[13]Shul'man, V.S., ‘Invariant subspaces of Volterra operators’, Funct. Anal. Appl. 18 (1984), 160–161.CrossRefGoogle Scholar
[14]Willis, G. A. ‘When the algebra generated by an operator is amenable’, J. Operator Theory 34 (1995), 239–249.Google Scholar