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

Quantum R\'enyi relative entropies on density spaces of $C^*$-algebras: their symmetries and their essential difference