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Finite dimensionality, nilpotents and quasinilpotents in Banach algebras

Published online by Cambridge University Press:  20 January 2009

J. Duncan  and
A. W. Tullo
J. Duncan
Affiliation:
University of Stirling
A. W. Tullo
Affiliation:
University of Stirling
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In this note we present some rather loosely connected results on Banach algebras together with some illustrative examples. We consider various conditions on a Banach algebra which imply that it is finite dimensional. We also consider conditions which imply the existence of non-zero nilpotents, and hence the existence of finite dimensional subalgebras. In the setting of Banach algebras quasinilpotents figure more prominently than nilpotents. We give an example of a non-commutative Banach algebra in which 0 is the only quasinilpotent; this resolves a problem of Hirschfeld and Zelazko (4).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 19 , Issue 1 , March 1974 , pp. 45 - 49
Copyright
Copyright © Edinburgh Mathematical Society 1974

References

REFERENCES

(1) Dixmier, J., Les C*-algèbres et leurs représentations (Gauthier-Villars, Paris, 1969).Google Scholar
(2) Dixon, P. G., Locally finite Banach algebras (preprint).Google Scholar
(3) Grabiner, S., Ranges of quasinilpotent operators, Illinois J. Math. 15 (1971), 150–152.Google Scholar
(4) Hirschfeld, R. A. and Zelazko, W., On spectral norm Banach algebras, Bull. Acad. Polon. Sci. 16 (1968), 195–199.Google Scholar
(5) Hirschfeld, R. A. and Rolewicz, S., A class of non-commutative Banach algebras without divisors of zero, Bull Acad. Polon. Sci. 17 (1969),751–753.Google Scholar
(6) Kaplansky, I., Ring isomorphisms of Banach algebras, Canad. J. Math.. 6 (1954), 374–381.Google Scholar
(7) Lepage, C., Sur quelques conditions entraînant la commutatitivité dans les algèbres de Banach, C. R. Acad. Sci. Paris Ser A-B 265 (1967), 235–237.Google Scholar
(8) Rickart, C. E., Banach Algebras (Van Nostrand, 1960).Google Scholar
(9) Zelazko, W., Concerning extensions of multiplicative linear functionals in Banach algebras, Studia Math. 31 (1968), 495–499.CrossRefGoogle Scholar