\(C^\ast\)-module Banach spaces (Q2818534)

The authors introduce the notion of a \(C^*\)-module Banach space as a generalization of that of a Banach space by allowing the ``norm'' to take its values in the positive cone of a \(C^*\)-algebra instead of the positive numbers. They then try to obtain some results analogous to elementary properties of Banach spaces. The critical points are as follows:NEWLINENEWLINE(1) The authors define ``\(a<b\)'' as ``\(a\leq b\) but \(a\neq b\)'' for arbitrary \(a, b \in A\), which is not possible in general since the usual order in \(A\) is defined on the real space of self-adjoint elements. On the other hand, a useful definition of ``\(a<b\)'' is ``\(a\leq b\) and \(b-a\) is invertible''.NEWLINENEWLINE(ii) The authors deal with \(C^*\)-algebras each of whose two elements have a least upper bound. The class of such \(C^*\)-algebras is rather narrow.NEWLINENEWLINE(iii) Implicitly, the authors assume that each two positive elements of a \(C^*\)-algebra are comparable with respect to the order \(\leq\) while this is `almost always' false.