## Illinois Journal of Mathematics

### Absolute-valuable Banach spaces

#### Abstract

Absolute-valuable Banach spaces are introduced as those Banach spaces which underlie complete absolute-valued algebras. Examples and counterexamples are given. It is proved that every Banach space can be isometrically enlarged to an absolute-valuable Banach space, which has the same density character as the given Banach space, and whose dual space is also absolute-valuable. It is also shown that every weakly countably determined Banach space different from $\mathbb{R}$ can be renormed in such a way that neither it nor its dual are absolute-valuable. Hereditarily indecomposable Banach spaces are examples of Banach spaces which cannot be renormed as absolute-valuable Banach spaces.

#### Article information

Source
Illinois J. Math., Volume 49, Number 1 (2005), 121-138.

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258138309

Digital Object Identifier
doi:10.1215/ijm/1258138309

Mathematical Reviews number (MathSciNet)
MR2157371

Zentralblatt MATH identifier
1084.46007

#### Citation

Becerra Guerrero, Julio; Moreno Galindo, Antonio; Rodríguez Palacios, Ángel. Absolute-valuable Banach spaces. Illinois J. Math. 49 (2005), no. 1, 121--138. doi:10.1215/ijm/1258138309. https://projecteuclid.org/euclid.ijm/1258138309