Abstract
The aim of this work is to study norm preserving extensions of completely positive and completely contractive maps on some spaces of fractions with values in a space of sesquilinear forms on a inner product space. The main result is stated in the case of non necessarily commutative C*-algebras with unit. Moreover, the main result is applied to characterize a "quaternionic" version of the operator moment problem. for this application, a special measure is introduced.