Wigner's unitary-antiunitary theorem in Hilbert C^*-modules
by
Boris Guljaš
Dept. of Mathematics, Zaagreb, Croatia
Coauthors: Damir Bakic

Wigner's classical unitary-antiunitary theorem states that each bijective function T: H --> H acting on a complex Hilbert space (H, ( ·, ·)) which satisfies |(Tx, Ty)|=|(x, y)|,  x, y in H, must be of the form Tx=j(x)Ux,  x in H, where U:H --> H is either unitary or antiunitary operator and j: H --> C is a phase function (i.e. its values are of modulus 1).

Let W be a Hilbert C^*-module over the C^*-algebra A=K(H) of all compact operators on a Hilbert space H, dimH > 1. We prove that any function T: W --> W which preserves the A-absolute value of the A-valued inner product is of the form Tv=j(v)Uv,  v in W, where j is a phase function and U is an A-linear isometry. The result generalizes Moln\' ar's extension of Wigner's unitary-antiunitary theorem.

An extension of Wigner's theorem to Hilbert C^*-modules over C^*-algebras containing K(H) will be also discussed.

Date received: June 26, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagt-54.