Hostname: page-component-7c8c6479df-94d59 Total loading time: 0 Render date: 2024-03-28T22:49:24.223Z Has data issue: false hasContentIssue false

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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