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

DOI:
https://doi.org/10.1090/S0002-9939-98-04358-5

MathSciNet review:
1451827

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Abstract | References | Similar Articles | Additional Information

Abstract: Contractive weak star continuous representations of the Fourier binest algebra (of Katavolos and Power) are shown to be completely contractive. The proof depends on the approximation of by semicrossed product algebras 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.

**1.**K.R.Davidson, V.I.Paulsen and S.C. Power, Tree algebras, semidiscreteness and dilation theory,*Proc London Math. Soc.*68 (1994), 178-202. MR**94m:47087****2.**K.R. Davidson and S.C. Power, Isometric automorphisms and homology for nonselfadjoint operator algebras,*Quart.Math.J.,*42 (1991), 271-292. MR**92h:47058****3.**M. Hamana, Injective envelopes of operator systems,*Publ. RIMS, Kyoto Univ.,*15 (1979), 773-785. MR**81h:46071****4.**T. Hoover, J. Peters and W. Wogen, Spectral Properties of semicrossed products, Houston J. Math. 19 (1993), 649-660. MR**94k:47067****5.**A. Katavolos and S.C. Power, The Fourier Binest Algebra,*Math. Proc. Camb. Phil. Soc.,*122 (1997), 525-539. CMP**97:17****6.**K-S Ling and P.S. Muhly, An automorphic form of Ando's theorem, J. Integral Equ. and Operator Th., 12 (1989),424-434. MR**90h:46103****7.**M.J. McAsey and P.S. Muhly, Representations of nonselfadjoint crossed products, Proc. London Math. Soc., 47 (1983), 128-144. MR**85a:46039****8.**V.I.Paulsen and S.C. Power, Lifting theorems for nest algebras,*J. Operator Th.,*20 (1988), 311-327. MR**90g:47010****9.**V.I. Paulsen, S.C. Power and J.O. Ward Semi-discreteness and dilation theory for nest algebras,*J. of Functional Anal.*80 (1988), 76-87. MR**89h:47064**

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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:
https://doi.org/10.1090/S0002-9939-98-04358-5

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