Isometries of the unitary groups in C*-algebras

We give a complete description of the structure of surjective isometries between the unitary groups of unital C*-algebras. While any surjective isometry between the unitary groups of von Neumann algebras can be extended to a real-linear Jordan *-isomorphism between the relevant von Neumann algebras, this is not the case for general unital C*-algebras. We show that the unitary groups of two C*-algebras are isomorphic as metric groups if and only if the C*-algebras are isomorphic in the sense that each of them can be decomposed as the direct sum of two C*-algebras with the first parts being linear *-algebra isomorphic and the second parts being conjugate-linear *-algebra isomorphic. We emphasize that in this paper by an isometry we merely mean a distance preserving transformation; we do not assume that it respects any algebraic operation.