Bures isometries between density spaces of C⁎-algebras

We describe the precise structure of all surjective ('a priori' nonlinear or, better say, nonaffine) Bures isometries between density spaces of C⁎-algebras equipped with faithful traces. It turns out that those maps are closely related to (linear) Jordan *-isomorphisms between the underlying algebras. Beside density spaces, we consider the problem also on the positive definite (and positive semidefinite) cones of C⁎-algebras.