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Automatic continuity with application to C*-algebras

Published online by Cambridge University Press:  24 October 2008

Angel Rodriguez Palacios
Affiliation:
Facultad de Ciencias, Departamento de Análisis Matemático, Universidad de Granada, 18071-Granada, Spain

Extract

The fact proved by Cleveland [4], that the topology of any (non-complete) algebra norm on a C*-algebra is stronger than the topology of the usual norm, is reencountered as a direct consequence of a theorem, which we prove in this note, stating that homomorphisms from certain non-complete normed (associative) algebras onto some semisimple Banach algebras are automatically continuous.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1990

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References

REFERENCES

[1]Aupetit, B.. The uniqueness of the complete norm topology in Banach algebras and Banach Jordan algebras. J. Fund. Anal. 47 (1982), 725.CrossRefGoogle Scholar
[2]Bonsall, F. F.. A minimal property of the norm in some Banach algebras. J. London Math. Soc. 29 (1954), 156164.CrossRefGoogle Scholar
[3]Bonsall, F. F. and Duncan, J.. Complete Normed Algebras (Springer-Verlag, 1973).CrossRefGoogle Scholar
[4]Cleveland, S. B.. Homomorphisms of non-commutative *-algebras. Pacific J. Math. 13 (1963), 10971109.CrossRefGoogle Scholar
[5]Esterle, J.. Norm d'algèbres minimales, topologie d'algèbre normée minimun sur certaines algèbres d'endomorphismes continus d'un espace normé. C. R. Acad. Sc. Paris Series A 277 (1973), 425427.Google Scholar
[6]Ransford, T. J.. A short proof of Johnson's uniqueness-of-norm theorem. Bull. London Math. Soc. (to appear).Google Scholar
[7]Rickart, C. E.. General Theory of Banach Algebras (Krieger, 1974).Google Scholar
[8]Sinclair, A. M.. Automatic Continuity of Linear Operators (Cambridge University Press, 1976).CrossRefGoogle Scholar
[9]Yood, B.. Homomorphisms on normed algebras. Pacific J. Math. 8 (1958), 373381.CrossRefGoogle Scholar