Wigner’s theorem in Hilbert 𝐶*-modules over 𝐶*-algebras of compact operators

Bakić, Damir; Guljaš, Boris

doi:10.1090/S0002-9939-02-06426-2

Wigner’s theorem in Hilbert $C^$-modules over $C^$-algebras of compact operators
HTML articles powered by AMS MathViewer

by Damir Bakić and Boris Guljaš PDF

Proc. Amer. Math. Soc. 130 (2002), 2343-2349 Request permission

Abstract:

Let $W$ be a Hilbert $C^*$-module over the $C^*$-algebra $\mathcal {A}\not = \boldsymbol {C}$ 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.

References

William Arveson, An invitation to $C^*$-algebras, Graduate Texts in Mathematics, No. 39, Springer-Verlag, New York-Heidelberg, 1976. MR 0512360
D. Bakić, B. Guljaš, Operators on Hilbert $H^*$-modules, accepted for publication in the Journal of Operator Theory.
D. Bakić, B. Guljaš, Hilbert $C^*$-modules over $C^*$-algebras of compact operators, accepted for publication in Acta Sci. Math. (Szeged).
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. MR 1316963, DOI 10.1017/S0017089500030378
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.
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
C. Lance, Hilbert $C^*$-modules, London Mat. Soc. Lecture Notes Series, 210, Cambridge University Press, Cambridge, 1995.
Lajos Molnár, An algebraic approach to Wigner’s unitary-antiunitary theorem, J. Austral. Math. Soc. Ser. A 65 (1998), no. 3, 354–369. MR 1660422
Lajos Molnár, A generalization of Wigner’s unitary-antiunitary theorem to Hilbert modules, J. Math. Phys. 40 (1999), no. 11, 5544–5554. MR 1722325, DOI 10.1063/1.533044
William L. Paschke, Inner product modules over $B^{\ast }$-algebras, Trans. Amer. Math. Soc. 182 (1973), 443–468. MR 355613, DOI 10.1090/S0002-9947-1973-0355613-0
Jürg Rätz, On Wigner’s theorem: remarks, complements, comments, and corollaries, Aequationes Math. 52 (1996), no. 1-2, 1–9. MR 1406619, DOI 10.1007/BF01818323
Marc A. Rieffel, Induced representations of $C^{\ast }$-algebras, Advances in Math. 13 (1974), 176–257. MR 353003, DOI 10.1016/0001-8708(74)90068-1
N. E. Wegge-Olsen, $K$-theory and $C^*$-algebras, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993. A friendly approach. MR 1222415
Albert Eagle, Series for all the roots of the equation $(z-a)^m=k(z-b)^n$, Amer. Math. Monthly 46 (1939), 425–428. MR 6, DOI 10.2307/2303037