Absolute-valuable Banach spaces (Q2567491)

In this interesting paper, the authors study the notion of `absolute valuability' for real or complex Banach spaces, analogous to the well-studied notion of absolute-valued algebras (see the recent survey article by \textit{A. Rodriguez} [in: Proceedings of the 1st international school ``Advanced courses of mathematical analysis 1'', Cádiz, Spain, September 22--27, 2002, 99--155 (2004; Zbl 1093.46022)]. We recall that a Banach space \(X\) is said to be absolute-valuable if there exists a bilinear map (product) \((x,y) \rightarrow xy\) on \(X\) such that \(\| xy\| =\| x\| \| y\| \). The authors show that this property is preserved under \(c_0\) and \(\ell^p\) sums. The space of compact and bounded operators between discrete \(\ell^p\) spaces has this property. However, for any infinite set \(\Gamma\), \(c(\Gamma)\) fails to be absolute-valuable. The same is the case for the \(\ell^1\)-direct sum of two non-zero almost smooth or almost strictly convex spaces. Turning to the isomorphic questions, the authors show that any real Banach space different from the real numbers that is weakly countably determined is isomorphic to a real Banach space \(X\) such that both \(X\) and \(X^\ast\) are not absolute-valuable. They also exhibit both separable and non-separable infinite-dimensional reflexive Banach spaces that are not isomorphic to any absolute-valuable Banach space.