Isometries and maps compatible with inverted Jordan triple products on groups
By the classical Mazur-Ulam theorem, every bijective isometry between two real normed spaces is linear up to a translation. The authors investigate generalisations of this theorem to certain subsets \(X\) of groups endowed with a metric or an even more general map \(d:X\times X \rightarrow [0,\infty)\). In Section 2, a general theorem (Theorem 2.4) on so-called \(d\)-preserving maps is proved (no group structure is required here). The proof is based on ideas of \textit{J. Väisälä}'s approach to the Mazur-Ulam theorem [Am. Math. Mon. 110, No. 7, 633--635 (2003; Zbl 1046.46017)]. In the subsequent sections, various applications of Theorem 2.4 are given. Section 3 deals with \(d\)-preserving maps on subsets \(X\) of groups which are closed under the so-called inverted Jordan triple product, i.e., \(yx^{-1}y\in X\) for all \(x,y\in X\). In Section 4, \(d\)-preserving maps between (multiplicative) groups of complex-valued continuous functions on compact Hausdorff spaces are studied. In particular (Corollary 4.5), if \(Y_1, Y_2\) are compact Hausdorff spaces and \(T:\exp iC_{\mathbb{R}}(Y_1) \rightarrow \exp iC_{\mathbb{R}}(Y_2)\) is a bijective isometry (with respect to the usual \(\sup\)-metric), then there is a homeomorphism \(\Phi: Y_2 \rightarrow Y_1\) and a decomposition of \(Y_2\) into clopen subsets \(Y_{21}, Y_{22}\) such that \(T(f)(y)=T(1)(y)f(\Phi(y))\) for \(y\in Y_{21}\) and \(T(f)(y)=T(1)(y)\overline{f(\Phi(y))}\) for \(y\in Y_{22}\) holds for every \(f\in \exp iC_{\mathbb{R}}(Y_1)\). This result also yields a characterisation of the bijective isometries on the torus group \(\mathbb{T}^n\) (Corollary 4.6). In the last section, a generalisation of the classical Mazur-Ulam theorem to certain commutative metric groups and some of its applications are presented.
metric spaces of positive invertible elements in \(C^\ast\)-algebras
scientific article; zbMATH DE number 6146578
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