Equality of entropies for states

We show that equality of Rényi’s or Tsallis’ entropies for two faithful normal states on a semifinite von Neumann algebra yields the existence of a ∗-isomorphism of the algebras generated by these states. Moreover, it is shown that this isomorphism can be extended to a normal completely positive map on the algebra. A similar result is shown for equality of the Segal or Rényi or Tsallis entropies of some perturbed states of the initial states on a finite von Neumann algebra.