Illinois Journal of Mathematics

On domination of inessential elements in ordered Banach algebras

D. Behrendt and H. Raubenheimer

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Abstract

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

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

Dates
First available in Project Euclid: 13 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258131111

Digital Object Identifier
doi:10.1215/ijm/1258131111

Mathematical Reviews number (MathSciNet)
MR2379731

Zentralblatt MATH identifier
1160.46029

Subjects
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

Citation

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. https://projecteuclid.org/euclid.ijm/1258131111


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