Hostname: page-component-77c89778f8-cnmwb Total loading time: 0 Render date: 2024-07-19T13:48:34.090Z Has data issue: false hasContentIssue false
  • English
  • Français

Annihilator and complemented banach*-algebras

Published online by Cambridge University Press:  09 April 2009

B. J. Tomiuk  and
Pak-Ken Wong
B. J. Tomiuk
Affiliation:
University of OttawaOttawa, Canada
Pak-Ken Wong
Affiliation:
University of OttawaOttawa, Canada
Rights & Permissions [Opens in a new window]

Extract

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.

The study of complemented Banach*-algebras taken up in [1] was confined mainly to B*-algebras. In the present paper we extend this study to (right) complemented Banach*-algebras in which x*x = 0 implies x = 0. We show that if A is such an algebra then every closed two-sided ideal of A is a *-ideal. Using this fact we obtain a structure theorem for A which states that if A is semi-simple then A can be expressed as a topological direct sum of minimal closed two sided ideals each of which is a complemented Banach*-algebra. It follows that A is an A*-algebra and is a dense subalgebra of a dual B*-algebra U, which is determined uniquely up to *-isomorphism.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 13 , Issue 1 , February 1972 , pp. 47 - 66
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Alexander, F. E. and Tomiuk, B. J., ‘Complemented Balgebras’, Trans. Amer. Math. Soc. 137 (1969), 459–480.Google Scholar
[2]Bonsall, F. F. and Goldie, A. W., ‘Annihilator algebras’, Proc. London Math. Soc. (3), 4 (1954), 154–167.Google Scholar
[3]Dixmer, J., Calgèbres et leurs représentations (Gauthier-Villars, Paris, 1964).Google Scholar
[4]Kakutani, I. and Mackey, G. W., ‘Ring and lattice characterizations of complex Hilbert space’, Bull. Amer. Math. Soc. 52 (1964), 727–733.Google Scholar
[5]Kaplansky, I., ‘The structure of certain operator algebras’, Trans. Amer. Math. Soc. 70 (1951), 219–255.CrossRefGoogle Scholar
[6]Ogasawara, T. and Yoshinaga, K., ‘Weakly completely continuous Banach-algebras’, J. Sci. Hiroshima Uni. Ser. A 18 (1954), 15–36.Google Scholar
[7]Ogasawara, T. and Yoshinaga, K., ‘A characterization of dual B-algebras’, J. Sci. Hiroshima Uni. Ser. A 18 (1954), 179–182.Google Scholar
[8]Rickart, C. E., General theory of Banach algebras (D. Van Nostrand, New York, 1960).Google Scholar
[9]Schatten, R., Norm ideals of completely continuous operators (Springer-Verlag, 1960).CrossRefGoogle Scholar
[10]Tomiuk, B. J., ‘Structure theory of complemented Banach algebras’, Canadian J. of Math. 14 (1962), 651–659.CrossRefGoogle Scholar