Isometries and Maps Compatible with Inverted Jordan Triple Products on Groups

Motivated by the famous Mazur-Ulam theorem in this paper we study algebraic properties of isometries between metric groups. We present some general results on so-called $d$-preserving maps between subsets of groups and apply them in several directions. We consider $d$-preserving maps on certain groups of continuous functions defined on compact Hausdorff spaces and describe the structure of isometries between groups of functions mapping into the circle group $\mathbb T$. Finally, we show a generalization of the Mazur-Ulam theorem for commutative groups and present a metric characterization of normed real-linear spaces among commutative metric groups.