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.

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