On isometries of symmetric products of metric spaces
type: Item
instance of: scholarly article
title: On isometries of symmetric products of metric spaces
zbMATH Open document ID: 1402.54023
author: Naotsugu Chinen
published in: Topology and its Applications
publication date: 2018-10-02T00:00:00Z
review text: For a metric space \(X\) and a positive integer \(n\), let \({F}_n (X)\) denote the metric space of all nonempty sets with at most \(n\) points endowed with the Hausdorff metric, and let \(\text{Iso}(X)\) denote the group of all isometries from \(X\) onto itself with the topology of pointwise convergence. Then each \(\phi \in \text{Iso}(X)\) defines an isometry \(\chi_n (\phi) \in \text{Iso}(F_n(X))\) by letting \(\chi_n(\phi)(A) =\phi (A)\) for \(A \in F_n(X)\), and this induces a canonical monomorphism \(\chi_n : \text{Iso} (X) \to \text{Iso}(F_n(X))\) which is also a topological embedding. For \(1 \leq p \leq \infty\) and \(q \in \mathbb{N} \cup \{\infty\}\), let \(\ell_p^q=\{(x_i)_{i=1}^q \in \mathbb{R}^q: \|(x_i)\|_p <\infty\}\), where \(\|(x_i)\|_p = (\sum_{i=1}^{q}|x_i|^p)^{1/p}\) if \(1\leq p <\infty\) and \(\|(x_i)\|_{\infty} = \sup \{ |x_i| : 1\leq i\leq q\}\). \textit{M. Borovikova} and \textit{Z. Ibragimov} [Comput. Methods Funct. Theory 9, No. 1, 255--268 (2009; Zbl 1156.54005)] proved that \(\chi_3 : \text{Iso} (\ell_2^1) \to \text{Iso}(F_3(\ell_2^1))\) is surjective (and hence a topological isomorphism). The author [Commentat. Math. Univ. Carol. 56, No. 2, 209--221 (2015; Zbl 1349.54032)] generalized this result by proving that \(\chi_n : \text{Iso} (\ell_2^q) \to \text{Iso}(F_n(\ell_2^q))\) is surjective for every \(n\geq 2\) and \(q \in \mathbb{N}\). On the other hand, \textit{C. Bandt} [Math. Nachr. 129, 175--183 (1986; Zbl 0609.54008)] showed that \(\chi_2 : \text{Iso} (\ell_\infty^2) \to \text{Iso}(F_2(\ell_\infty^2))\) is not surjective. In this paper, the author proves the following theorems: (1) If \(p \in \{1,\infty\}\), then \(\chi_2 : \text{Iso} (\ell_p^2) \to \text{Iso}(F_2(\ell_p^2))\) is not surjective and \(\text{Iso}(F_2(\ell_p^2))\) is topologically isomorphic to \(\mathbb{Z}_2 \times \text{Iso} (\ell_p^2) \). (2) If \(3 \leq q < \infty\), then \(\chi_2 : \text{Iso} (\ell_\infty^q) \to \text{Iso}(F_2(\ell_\infty^q))\) is not surjectiv…
reviewed by: Takamitsu Yamauchi
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 22F50
Mathematics Subject Classification ID: 30C65
Mathematics Subject Classification ID: 30L10
Mathematics Subject Classification ID: 22A05
zbMATH DE Number: 6946116
zbMATH Keywords: isometry
zbMATH Keywords: symmetric product
zbMATH Keywords: topological groups
zbMATH Keywords: semidirect product
zbMATH Keywords: normed space
zbMATH Keywords: \(\ell_p^q\)
MaRDI profile type: Publication
full work available at URL: j.topol.2018.08.006
OpenAlex ID: W2886664397
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DOI: 10.1016/J.TOPOL.2018.08.006
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description: scientific article; zbMATH DE number 6946116