On representations of partial ^*-algebras based on B-weights

A generalization of the GNS-representation is investigated that represents partial ^*-algebras as systems of operators acting on a partial inner product space (PIP-space). It is based on possibly indefinite B-weights which are closely related to the positive B-weights introduced by J.-P. Antoine, Y. Soulet and C. Trapani. Some additional assumptions had to be made in order to guarantee the GNS-construction. Different partial products of operators on a PIP-space are considered which allow the GNS-construction under suitable conditions. Several examples illustrate the argumentation and indicate inherent problems.