By a generalized distance measure on a set $X$ we mean a nonnegative real-valued function $d$ on $X\times X$ such that $d(x,y)=0$ holds precisely when $x=y$, $x,y\in X$. Metrics are trivial examples but large classes of matrix divergences, relative entropies, etc also provide important examples of such measures.
Due to some abstract versions of Mazur-Ulam theorem, it turns out in many cases that surjective transformations which leave a generalized distance measure invariant (in particular, isometries) are closely related to certain algebraic transformations (sort of isomorphisms) called Jordan triple maps.
In this talk we extend in a way and unify our previous investigations on the connections between the isometries and the isomorphisms of non-linear spaces of matrices and operators. We focus on two sorts of structures, namely, positive definite cones and unitary groups in operator algebras ($C^*$-algebras, or von Neumann algebras, or von Neumann factors). We describe the general form of (continuous) Jordan triple maps on them and then, applying the above mentioned abstract Mazur-Ulam theorems, we determine the surjective transformations that respect generalized distance measures belonging to a broad class.
Former results on the form of surjective isometries of the positive definite cone equipped with the Thompson part metric and those of the unitary group relative to the operator norm are simple consequences of these new results.