On representations of partial *-algebras based on \(\mathfrak B\)-weights (Q5926106)

Summary: 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 \textit{J.-P. Antoine}, \textit{Y. Soulet} and \textit{C. Trapani} [J. Math. Anal. Appl. 192, No. 3, 920-941 (1995; Zbl 0888.46034)]. 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.