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On Partially Ordering Operator Algebras

Published online by Cambridge University Press:  20 November 2018

Ralph Demarr
Ralph Demarr*
Affiliation:
University of Washington
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In this paper, we consider linear spaces and algebras with real scalars. It is well known that if X is a Banach space and is the set of all bounded linear operators which map X into itself, then is a Banach algebra. In this paper we shall show that can be partially ordered so that it becomes a partially ordered algebra in which norm convergence is equivalent to order convergence. This motivates a study of Banach algebras of operators in which one uses the order structure to obtain various results. In addition, it encourages a study of partially ordered algebras in general, since our result shows that among such algebras one finds all real Banach algebras of operators. Of course, there are many other real algebras which are naturally partially ordered and which have been studied from that point of view.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 636 - 643
Copyright
Copyright © Canadian Mathematical Society 1967

References

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