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Wigner’s theorem in Hilbert $$C^*$$-modules over $$C^*$$-algebras of compact operators. (English) Zbl 1067.46052
The notion of Hilbert $$C^{*}$$-module is a generalization of the notion of Hilbert space, by allowing the inner product to take values in a $$C^{*}$$-algebra.
In the paper under review, the authors show that if $$W$$ is a Hilbert $$C^{*}$$-module over the $$C^{*}$$-algebra of all compact operators on a Hilbert space $$H$$ with $$\dim H>1$$ and $$T:W\rightarrow W$$ is a function satisfying $$| \langle Tv,Tw\rangle| =|\langle v,w\rangle|$$ for all $$v,w\in W$$, then there exist an adjointable map $$U:W\rightarrow W$$ which is an isometry (or equivalently $$U^{*}U=I$$) and there is a phase function $$\phi:W\rightarrow \mathbb C$$ (i.e., a function whose values are of modulus $$1$$) such that $$Tv=\phi(v)Uv$$ for all $$v\in W$$.
This result generalizes L. Molnár’s extension [J. Math. Phys. 40, No.11, 5544-5554 (1999; Zbl 0953.46030)]) of Wigner’s classical unitary-antiunitary theorem [E. Wigner [“Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren” (Die Wissenschaft 85, F. Vieweg & Sohn, Braunschweig) (1931; JFM 57.1578.03)]).
The authors conclude the paper with the conjecture that the main result of the paper is true for Hilbert modules over concrete $$C^{*}$$-algebras which contain the ideal of all compact operators.

##### MSC:
 46L08 $$C^*$$-modules 46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) 46C50 Generalizations of inner products (semi-inner products, partial inner products, etc.) 39B42 Matrix and operator functional equations 47J05 Equations involving nonlinear operators (general)
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##### References:
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