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Zeitschrift für Analysis und ihre Anwendungen


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Volume 19, Issue 3, 2000, pp. 623–638
DOI: 10.4171/ZAA/971

Published online: 2000-09-30

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

Klaus-Detlef Kürsten[1] and Elmar Wagner[2]

(1) Universität Leipzig, Germany
(2) Universität Leipzig, Germany

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.

Keywords: Partial algebras, $GNS$-construction, weights, partial inner product spaces

Kürsten Klaus-Detlef, Wagner Elmar: On Representations of Partial *-Algebras Based on $\mathfrak B$-Weights. Z. Anal. Anwend. 19 (2000), 623-638. doi: 10.4171/ZAA/971