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Isometries of Hilbert $C^*$-modules


Author: Baruch Solel
Journal: Trans. Amer. Math. Soc. 353 (2001), 4637-4660
MSC (2000): Primary 46L08
DOI: https://doi.org/10.1090/S0002-9947-01-02874-4
Published electronically: July 3, 2001
MathSciNet review: 1851186
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Abstract:

Let $X$ and $Y$ be right, full, Hilbert $C^*$-modules over the algebras $A$ and $B$ respectively and let $T:X\to Y$ be a linear surjective isometry. Then $T$ can be extended to an isometry of the linking algebras. $T$ then is a sum of two maps: a (bi-)module map (which is completely isometric and preserves the inner product) and a map that reverses the (bi-)module actions. If $A$(or $B$) is a factor von Neumann algebra, then every isometry $T:X\to Y$ is either a (bi-)module map or reverses the (bi-)module actions.

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Additional Information

Baruch Solel
Affiliation: Department of Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel
Email: mabaruch@techunix.technion.ac.il

DOI: https://doi.org/10.1090/S0002-9947-01-02874-4
Received by editor(s): June 26, 2000
Published electronically: July 3, 2001
Additional Notes: Supported by Technion V.P.R. Fund–Steigman Research Fund, Technion V.P.R. Fund–Fund for the Promotion of Sponsored Research and the Fund for the Promotion of Research at the Technion.
Article copyright: © Copyright 2001 American Mathematical Society

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