On Representations of Partial *-Algebras Based on $\mathfrak{B}$-Weights

Abstract

A generalization of the $GNS$-representation is investigated that represents partial *-algebras as systems of operators acting on a partial inner product space (i.e., $PIP$-space). It is based on possibly indefinite $\mathfrak{B}$-weights which are closely related to the positive $\mathfrak{B}$-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.