Homogeneity Properties of Some ℓ1-Spaces

Let M be a family of subspaces of a metric space X. The space X is M-homogeneous if any isometry between two subspaces in M can be extended to an isometry of X. In the case when M is the family of all subspaces of X, the space X is said to be fully homogeneous. We show that the space Ln, where L is a fully homogeneous subspace of ℝ, is R-homogeneous where R is the family of subspaces that are products of subsets of L.