Homogeneity properties of some \(\ell_1\)-spaces
Let \(\mathcal M\) be a family of subspaces of a metric space \(X\). The space \(X\) is \(\mathcal M\)-homogeneous if any isometry of two subspaces in \(\mathcal M\) can be extended to an isometry of \(X\). The author calls \(X\) fully homogeneous when \(\mathcal M\) is the family of all subspaces of \(X\). It is shown that a fully homogeneous subspace \(L\) of \(\mathbb R\) is either a two-point subspace or a subspace isometric to a subgroup of the additive group \(\mathbb R\), and that the space \(L^n\), where \(L\) is a fully homogeneous subspace of \(\mathbb R\), is \(\mathcal R\)-homogeneous, where \(\mathcal R\) is the family of subspaces that are products of subsets of \(L\).
scientific article; zbMATH DE number 5013772