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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Completely contractive representations
for some doubly generated
antisymmetric operator algebras


Author: S. C. Power
Journal: Proc. Amer. Math. Soc. 126 (1998), 2355-2359
MSC (1991): Primary 46K50
MathSciNet review: 1451827
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Abstract: Contractive weak star continuous representations of the Fourier binest algebra $\mathcal A$ (of Katavolos and Power) are shown to be completely contractive. The proof depends on the approximation of $\mathcal A$ by semicrossed product algebras $A(\mathbb D)\times\mathbb Z_+$ and on the complete contractivity of contractive representations of such algebras. The latter result is obtained by two applications of the Sz.-Nagy-Foias lifting theorem. In the presence of an approximate identity of compact operators it is shown that an automorphism of a general weakly closed operator algebra is necessarily continuous for the weak star topology and leaves invariant the subalgebra of compact operators. This fact and the main result are used to show that isometric automorphisms of the Fourier binest algebra are unitarily implemented.


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Additional Information

S. C. Power
Affiliation: Department of Mathematics and Statistics, Lancaster University, LA1 4YF, England
Email: s.power@lancaster.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04358-5
PII: S 0002-9939(98)04358-5
Received by editor(s): December 18, 1995
Received by editor(s) in revised form: February 22, 1996, March 4, 1996, and January 21, 1997
Additional Notes: Partially supported by a NATO Collaborative Research Grant
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society