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Annihilator and complemented banach*-algebras

Published online by Cambridge University Press:  09 April 2009

B. J. Tomiuk  and
Pak-Ken Wong
B. J. Tomiuk
Affiliation:
University of OttawaOttawa, Canada
Pak-Ken Wong
Affiliation:
University of OttawaOttawa, Canada
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The study of complemented Banach*-algebras taken up in [1] was confined mainly to B*-algebras. In the present paper we extend this study to (right) complemented Banach*-algebras in which x*x = 0 implies x = 0. We show that if A is such an algebra then every closed two-sided ideal of A is a *-ideal. Using this fact we obtain a structure theorem for A which states that if A is semi-simple then A can be expressed as a topological direct sum of minimal closed two sided ideals each of which is a complemented Banach*-algebra. It follows that A is an A*-algebra and is a dense subalgebra of a dual B*-algebra U, which is determined uniquely up to *-isomorphism.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 13 , Issue 1 , February 1972 , pp. 47 - 66
Copyright
Copyright © Australian Mathematical Society 1972

References

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