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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.

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)
Full Text: DOI
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