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Strictly real Banach algebras

Published online by Cambridge University Press:  17 April 2009

John Boris Miller
John Boris Miller
Affiliation:
Department of Mathematics, Monash University Clayton, Victoria 3168, Australia
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A complex Banach algebra is a complexification of a real Banach algebra if and only if it carries a conjugation operator. We prove a uniqueness theorem concerning strictly real selfconjugate subalgebras of a given complex algebra. An example is given of a complex Banach algebra carrying two distinct but commuting conjugations, whose selfconjugate subalgebras are both strictly real. The class of strictly real Banach algebras is shown to be a variety, and the manner of their generation by suitable elements is proved. A corollary describes some strictly real subalgebras in Hermitian Banach star algebras, including C* algebras.

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 47 , Issue 3 , June 1993 , pp. 505 - 519
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Bonsall, F.F. and Duncan, J., Complete normed algebras (Springer-Verlag, Berlin, Heidelberg, New York, 1973).CrossRefGoogle Scholar
[2]Ingelstam, L., ‘Real Banach algebras’, Ark. Mat. 5 (1964), 239–270.Google Scholar
[3]Kadison, R.V. and Ringrose, J.R., Fundamentals of the theory of operator algebras, Vol 1. (Academic Press, New York, 1983).Google Scholar
[4]Kaplansky, I., ‘Normed algebras’, Duke Math. J. 16 (1949), 399–418.Google Scholar
[5]Miller, J.B., ‘The natural ordering on a strictly real Banach algebra’, Math. Proc. Cambridge Philos. Soc. 107 (1990), 539–556.CrossRefGoogle Scholar
[6]Miller, J.B., ‘Analysis in strictly real Banach algebras’, Monash Univ. Analysis Paper No.70 (Monash University, Clayton, Australia).Google Scholar
[7]Rickart, C., General theory of Banach algebras (Van Nostrand, Princeton, 1960).Google Scholar