Journal of Operator Theory

Volume 44, Issue 1, Summer 2000 pp. 151-205.

Regular ${C^*}$-valued weights

Authors: Johan Kustermans
Author institution: Department of Mathematics, University College Cork, Western Road, Cork, Ireland

Summary: We introduce the notion of a $C^*$-valued weight between two $C^*$-algebras as a generalization of an ordinary weight on a $C^*$-algebra and as a $C^*$-version of operator valued weights on von Neumann algebras. Also, some form of lower semi-continuity will be discussed together with an extension to the multiplier algebra. A strong but useful condition for $C^*$-valued weights, the so-called regularity, is introduced. At the same time, we propose a construction procedure for such regular $C^*$-valued weights. This construction procedure will be used to define the tensor product of regular $C^*$-valued weights.

Keywords: $C^*$-algebra, Hilbert $C^*$-module, weight, ${\rm KSGNS}$-construction

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