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.