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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Wigner's theorem in Hilbert $C^*$-modules over $C^*$-algebras of compact operators


Authors: Damir Bakic and Boris Guljas
Journal: Proc. Amer. Math. Soc. 130 (2002), 2343-2349
MSC (1991): Primary 46C05, 46C50; Secondary 39B42, 47J05
Published electronically: March 8, 2002
MathSciNet review: 1897459
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Abstract: Let $W$ be a Hilbert $C^*$-module over the $C^*$-algebra $\mathcal{A}\not = \boldsymbol{\mathit{C}}$of all compact operators on a Hilbert space. It is proved that any function $T: W \rightarrow W$ which preserves the absolute value of the ${\mathcal A}$-valued inner product is of the form $Tv=\varphi(v)Uv,\, v \in W$, where $\varphi$ is a phase function and $U$ is an ${\mathcal A}$-linear isometry. The result generalizes Molnár's extension of Wigner's classical unitary-antiunitary theorem.


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Additional Information

Damir Bakic
Affiliation: Department of Mathematics, University of Zagreb, Bijenička c. 30, 10000 Zagreb, Croatia
Email: bakic@math.hr

Boris Guljas
Affiliation: Department of Mathematics, University of Zagreb, Bijenička c. 30, 10000 Zagreb, Croatia
Email: guljas@math.hr

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06426-2
PII: S 0002-9939(02)06426-2
Keywords: $C^*$-algebra, Hilbert $C^*$-module, compact operator, Wigner's theorem
Received by editor(s): October 2, 2000
Received by editor(s) in revised form: March 12, 2001
Published electronically: March 8, 2002
Communicated by: David R. Larson
Article copyright: © Copyright 2002 American Mathematical Society