Isometries of the unitary groups and Thompson isometries of the spaces of invertible positive elements in -algebras

We show that the existence of a surjective isometry (which is merely a distance preserving map) between the unitary groups of unital C ∗ -algebras implies the existence of a Jordan *-isomorphism between the algebras. In the case of von Neumann algebras we describe the structure of those isometries showing that any of them is extendible to a real linear Jordan *-isomorphism between the underlying algebras multiplied by a fixed unitary element. We present a result of similar spirit for the surjective Thompson isometries between the spaces of all invertible positive elements in unital C ∗ -algebras.