On Tsallis relative entropies and their preservers on positive cones in operator algebras and in matrix algebras

In this paper, we are mainly concerned with Tsallis relative entropies and Tsallis operator relative entropies on positive cones in $C^{⁎}$-algebras but also consider their limit cases, the Umegaki relative entropy and the relative operator entropy. Motivated by Wigner's result on quantum mechanical symmetry transformations, we describe the surjective transformations on positive cones that preserve any of the corresponding numerical relative entropies. In the case of matrix algebras, we present related results where the surjectivity assumption is dropped. We will see that, although the numerical quantities under consideration do not suggest any sort of linearity, their preservers are intimately connected to the most fundamental linear and algebraic isomorphisms of the underlying full algebras.