Illinois Journal of Mathematics

On domination of inessential elements in ordered Banach algebras

D. Behrendt and H. Raubenheimer

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If $A$ is an ordered Banach algebra ordered by an algebra cone $C$, then we reference the following problem as the `domination problem': If $0\leq a\leq b$ and $b$ has a certain property, then does $a$ inherit this property? We extend the analysis of this problem in the setting of radical elements and introduce it for inessential, rank one and finite elements. We also introduce the class of $r$-inessential operators on Banach lattices and prove that if $S$ and $T$ are operators on a Banach lattice $E$ such that $0\leq S\leq T$ and $T$ is $r$-inessential then $S$ is also $r$-inessential.

Article information

Illinois J. Math., Volume 51, Number 3 (2007), 927-936.

First available in Project Euclid: 13 November 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46H05: General theory of topological algebras
Secondary: 46B40: Ordered normed spaces [See also 46A40, 46B42] 46H10: Ideals and subalgebras 47B60: Operators on ordered spaces


Behrendt, D.; Raubenheimer, H. On domination of inessential elements in ordered Banach algebras. Illinois J. Math. 51 (2007), no. 3, 927--936. doi:10.1215/ijm/1258131111.

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