Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-25T05:47:20.106Z Has data issue: false hasContentIssue false
  • English
  • Français

Reflexive Representations and Banach C*-Modules

Published online by Cambridge University Press:  20 November 2018

Don Hadwin  and
Mehmet Orhon
Don Hadwin
Affiliation:
Mathematics Department, University of New Hampshire, Durham, NH 03824, e-mail: [email protected]
Mehmet Orhon
Affiliation:
RR2 Box 5465, Union, ME 04862
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Suppose A is a unital C*-algebra and m:A → B(X) is unital bounded algebra homomorphism where B(X) is the algebra of all operators on a Banach space X. When X is a Hilbert space, a problem of Kadison [9] asks whether m is similar to a *-homomorphism. Haagerup [5] has shown that the answer is positive when m(A) has a cyclic vector or whenever m is completely bounded. We use this to show m(A) is reflexive (Alg Lat m(A) = m(A)−sot) whenever X is a Hilbert space. Our main result is that whenever A is a separable GCR C*-algebra and X is a reflexive Banach space, then m(A) is reflexive.

Keywords

Primary: 47D30Secondary46L99
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 40 , Issue 4 , 01 December 1997 , pp. 443 - 447
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Abramovitch, Y. A., Arenson, E. L., and Kitover, A. K., Banach C(K)-modules and operators preserving disjointness. Pitman Research Notes, 277, Wiley, 1992, New York.Google Scholar
2. Akemann, C., Dodds, P. G., and Gamlen, J. L. B., Weak compactness in the dual space of a C*-algebra. J. Funct. Anal. 10 (1972), 446–450.Google Scholar
3. Bade, W. G., On Boolean algebras of projections and algebras of operators. Trans. Amer. Math. Soc. 80 (1955), 345–360.Google Scholar
4. Dieudonné, J., Sur la bicommutante d’un algèbre d’operateurs. Portugaliae Math. 14 (1955), 35–38.Google Scholar
5. Haagerup, U., Solution of the similarity problem for cyclic representations of C*-algebras. Ann. of Math. 118 (1983), 215–240.Google Scholar
6. Hadwin, D. W., An asymptotic double commutant theorem for C*-algebras. Trans. Amer. Math. Soc. 244 (1978), 273–297.Google Scholar
7. Hadwin, D. W. and Orhon, M., Reflexivity and approximate reflexivity for Boolean algebras of projections. J. Funct. Anal. 87 (1989), 348–358.Google Scholar
8. Hadwin, D. W., A noncommutative theory of Bade functionals. Glasgow Math. J, 33 (1991), 73–81.Google Scholar
9. Kadison, R., On the orthogonalization of operator representations. Amer. J.Math. 77 (1955), 600–620.Google Scholar
10. Orhon, M., Boolean algebras of commuting projections. Math. Z. 183 (1983), 531–537.Google Scholar
11. Pedersen, G., C*-algebras and their automorphism groups. London Math. Soc. Monographs 14(1979), London.Google Scholar
12. Pełczynski, A., Projections in certain Banach spaces. Stud. Math. 19 (1960), 209–228.Google Scholar