Maps on the positive definite cone of a C⁎-algebra preserving certain quasi-entropies

We describe the structure of those bijective maps on the cone of all positive invertible elements of a C⁎C⁎-algebra with a normalized faithful trace which preserve certain kinds of quasi-entropy. It is shown that essentially any such map is equal to a Jordan *-isomorphism of the underlying algebra multiplied by a central positive invertible element.