Isometries and maps compatible with inverted Jordan triple products on groups
type: Item
instance of: scholarly article
title: Isometries and maps compatible with inverted Jordan triple products on groups
zbMATH Open document ID: 1271.46010
author: Osamu Hatori
author: Go Hirasawa
author: T. Miura
author: Lajos Molnár
published in: Tokyo Journal of Mathematics
publication date: 2013-03-19T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.tjm/1358951327
review text: 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.
reviewed by: Jan-David Hardtke
Mathematics Subject Classification ID: 46B04
Mathematics Subject Classification ID: 46J10
Mathematics Subject Classification ID: 20K30
zbMATH DE Number: 6146578
zbMATH Keywords: Mazur-Ulam theorem
zbMATH Keywords: \(d\)-preserving maps
zbMATH Keywords: isometries on groups
zbMATH Keywords: metric spaces of positive invertible elements in \(C^\ast\)-algebras
zbMATH Keywords: unitary groups
zbMATH Keywords: groups in function spaces
MaRDI profile type: Publication
OpenAlex ID: W2086580230
cites work: Q5615632
cites work: Isometries between groups of invertible elements in Banach algebras
cites work: Algebraic properties of isometries between groups of invertible elements in Banach algebras
cites work: Reflections and a generalization of the Mazur-Ulam theorem
cites work: Isometries of the unitary group
cites work: Q4333558
cites work: Divisibly norm-preserving maps between commutative Banach algebras
cites work: Thompson isometries of the space of invertible positive operators
cites work: Thompson isometries on positive operators: The 2-dimensional case
cites work: On the groups isomorphism of unitary groups in \(AW^*\)-algebras
cites work: Maps which preserve equality of distance
cites work: A Proof of the Mazur-Ulam Theorem
cites work: Q3625986
DOI: 10.3836/TJM/1358951327
description: scientific article; zbMATH DE number 6146578