engleski

Wigner's theorem in a class of Hilbert C*-modules

Let 𝐻 be a complex Hilbert space, dim H>1, and let 𝒜⊆B(H) be a 𝐶*-algebra such that the ideal K(H) of all compact operators on 𝐻 is contained in A. Let 𝑋 be a Hilbert 𝐶*-module over A. We prove that any function F:X→X which preserves the absolute value of the A-valued inner product on 𝑋 is of the form F(x)=φ(x)Ux, x∈X, where φ is a phase function and 𝑈 is an A-linear isometry. The result generalizes Wigner’s classical unitary-antiunitary theorem and its extension to Hilbert K(H) -modules.

Hilbert C*-module

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