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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).

Information

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

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