Journal of Operator Theory

Volume 48, Issue 3, Supplementary 2002 pp. 621-632.

Hilbert $C^*$-modules with a predual

Authors: Jurgen Schweizer
Author institution: Mathematisches Institut, Universitaet Tubingen, Auf der Morgenstelle 10, 72076 Tubingen, Germany

Summary: We extend Sakai's characterization of von Neumann algebras to the context of Hilbert $C^*$-modules. If $A,B$ are $C^*$-algebras an d $X$ is a full Hilbert $A$-$B$-bimodule possessing a predual such that left, respectively right, multiplications are weak*-continuous, then $\text{M}(A)$ and $\text{M}(B)$ are $W^*$-algebras, the predual is unique, and $ X$ is selfdual in the sense of Paschke. For unital $A,B$ the above continuity requirement is automatic. We determine the dual Banach space $X^*$ of a Hilbert $A$-$B$-bimodule $X$ and show that Paschke's selfdual completion of $X$ is isomorphic to the bidual $X^{**}$, which is a Hilbert $C^*$-module in a natural way. We conclude with a new approach to multipliers of Hilbert $C^*$-bimodules.

Keywords: Hilbert $W^*$-module, Hilbert $C^*$-module, correspondence

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