  We extend the definitions of different types of quantum R\'enyi relative
entropy from the finite dimensional setting of density matrices to density
spaces of $C^*$-algebras. We show that those quantities (which trivially
coincide in the classical commutative case) are essentially different on
non-commutative algebras in the sense that none of them can be transformed to
another one by any surjective transformation between density spaces. Besides,
we determine the symmetry groups of density spaces corresponding to each of
those quantum R\'enyi relative entropies and find that they are identical.
Similar results concerning the Umegaki and the Belavkin-Staszewksi relative
entropies are also presented.
