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A sharpening of the Berkson-Glickfeld theorem

Published online by Cambridge University Press:  20 January 2009

Zoltan Magyar
Zoltan Magyar
Affiliation:
Budapest, XXI Kiss J.a.u. 57. VIII/25, 1211, Hungary
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It is known that if in a Banach*-algebra with unit the following holds:

then it is a C*-algebra (see [3]).

We shall show that the above theorem can be sharpened in the following way: we replace the submultiplicativity of the norm by the weaker assumption

Observe that under this assumption even the existence of exp(ih) is not at all obvious, but it will be proved to be true below. Our main result is Theorem 2 which depends on Theorem 1. Our last remark is the equivalent-norm-version of the statement.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 26 , Issue 2 , June 1983 , pp. 275 - 278
Copyright
Copyright © Edinburgh Mathematical Society 1983

References

REFERENCES

1.Bonsall, F. F. and Duncan, J., Complete Normed Algebras (Erg. Math. Bd. 80, Springer, 1973).CrossRefGoogle Scholar
2.Ptak, V., Banach algebras with involution, Manuscripta Math. 6 (1972), 245–290.CrossRefGoogle Scholar
3.Glickfeld, B. W., A metric characterization of C(X) and its generalization to C*-algebras, Illinois J. Math. 10 (1966), 547–566.CrossRefGoogle Scholar