On isometries of symmetric products of metric spaces
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 surjective and \(\text{Iso}(F_2(\ell_\infty^q))\) is topologically isomorphic to \(\prod_{i=1}^{q-1}(\mathbb{Z}_2)_i \rtimes \text{Iso} (\ell_\infty^q) \). (3) In other cases except \((n,p,q) \in \mathbb{N}_{\geq 2} \times \{1,\infty\} \times \{\infty\}\), the canonical monomorphism \(\chi_n : \text{Iso} (\ell_p^q) \to \text{Iso}(F_n(\ell_p^q))\) is surjective.
scientific article; zbMATH DE number 6946116
scientific article