×

zbMATH — the first resource for mathematics

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)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] William Arveson, An invitation to \?*-algebras, Springer-Verlag, New York-Heidelberg, 1976. Graduate Texts in Mathematics, No. 39. · Zbl 0344.46123
[2] D. Bakic, B. Guljas, Operators on Hilbert \(H^*\)-modules, accepted for publication in the Journal of Operator Theory.
[3] D. Bakic, B. Guljas, Hilbert \(C^*\)-modules over \(C^*\)-algebras of compact operators, accepted for publication in Acta Sci. Math. (Szeged). · Zbl 1067.46052
[4] M. Cabrera, J. Martínez, and A. Rodríguez, Hilbert modules revisited: orthonormal bases and Hilbert-Schmidt operators, Glasgow Math. J. 37 (1995), no. 1, 45 – 54. · Zbl 0833.46037
[5] M. Frank, D. R. Larson, Frames in Hilbert \(C^*\)-modules and \(C^*\)-algebras, preprint, University of Houston, Houston, and Texas A&M University, College Station, Texas, USA, 1998.
[6] I. Kaplansky, Modules over operator algebras, Amer. J. Math. 75(1953), 839-853. · Zbl 0051.09101
[7] C. Lance, Hilbert \(C^*\)-modules, London Mat. Soc. Lecture Notes Series, 210, Cambridge University Press, Cambridge, 1995.
[8] Lajos Molnár, An algebraic approach to Wigner’s unitary-antiunitary theorem, J. Austral. Math. Soc. Ser. A 65 (1998), no. 3, 354 – 369. · Zbl 0943.46033
[9] Lajos Molnár, A generalization of Wigner’s unitary-antiunitary theorem to Hilbert modules, J. Math. Phys. 40 (1999), no. 11, 5544 – 5554. · Zbl 0953.46030
[10] William L. Paschke, Inner product modules over \?*-algebras, Trans. Amer. Math. Soc. 182 (1973), 443 – 468. · Zbl 0239.46062
[11] Jürg Rätz, On Wigner’s theorem: remarks, complements, comments, and corollaries, Aequationes Math. 52 (1996), no. 1-2, 1 – 9. · Zbl 0860.39033
[12] Marc A. Rieffel, Induced representations of \?*-algebras, Advances in Math. 13 (1974), 176 – 257. · Zbl 0284.46040
[13] N. E. Wegge-Olsen, \?-theory and \?*-algebras, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993. A friendly approach.
[14] E. Wigner, Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren, Vieweg, Braunschweig, 1931. (reprint) · JFM 57.1578.03
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.