Pre-Hilbert modules, normed modules and the parallelogram law

The concept of pre-Hilbert $C^*$-module generalizes the concept of pre-Hilbert (inner product) space. A normed $C^*$-module can be analogously introduced as a generalization of a normed space (by equipping a module over a $C^*$-algebra with a map that obeys the same axioms as the vector space norm but with values in a $C^*$-algebra). The aim of this talk is to show that the parallelogram law holds in every normed module over a $C^*$-algebra $A$ without nonzero commutative closed two-sided ideals and that this implies that the class of normed $A$-modules coincides with the class of pre-Hilbert $A$-modules.