A generalization of the -representation is investigated that represents partial *-algebras as systems of operators acting on a partial inner product space (i.e., -space). It is based on possibly indefinite -weights which are closely related to the positive -weights introduced by J.-P. Antoine, Y. Soulet and C. Trapani. Some additional assumptions had to be made in order to guarantee the -construction. Different partial products of operators on a -space are considered which allow the -construction under suitable conditions. Several examples illustrate the argumentation and indicate inherent problems.
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Klaus-Detlef Kürsten, Elmar Wagner, On Representations of Partial *-Algebras Based on -Weights. Z. Anal. Anwend. 19 (2000), no. 3, pp. 623–638