Illinois Journal of Mathematics

Absolute-valuable Banach spaces

Julio Becerra Guerrero, Antonio Moreno Galindo, and Ángel Rodríguez Palacios

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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

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

First available in Project Euclid: 13 November 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46B04: Isometric theory of Banach spaces
Secondary: 46B03: Isomorphic theory (including renorming) of Banach spaces 46H70: Nonassociative topological algebras [See also 46K70, 46L70]


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.

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