## Illinois Journal of Mathematics

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

### On domination of inessential elements in ordered Banach algebras

D. Behrendt and H. Raubenheimer

#### 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