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A note on Tingley’s problem and Wigner’s theorem in the unit sphere of \(\mathcal{L}^{\infty} (\Gamma)\)-type spaces. (English) Zbl 1489.46016

Summary: Suppose that \(f : S_X \rightarrow S_Y\) is a surjective map between the unit spheres of two real \(\mathcal{L}^{\infty} (\Gamma)\)-type spaces \(X\) and \(Y\) satisfying the following equation \[ \{\Vert f(x)+f(y)\Vert, \Vert f(x)-f(y)\Vert\} = \{\Vert x+y\Vert, \Vert x-y\Vert\} \quad (x,y \in S_X). \] We show that such a mapping \(f\) is phase equivalent to an isometry, i.e., there exists a function \(\varepsilon : S_X \rightarrow \{ -1, 1\}\) such that \(\varepsilon f\) is an isometry. We further show that this isometry is the restriction of a linear isometry between the whole spaces. These results can be seen as a combination of Tingley’s problem and Wigner’s theorem for \(\mathcal{L}^{\infty} (\Gamma)\)-type spaces.

MSC:

46B04 Isometric theory of Banach spaces
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