Spherical isometries of finite dimensional C⁎-algebras

In this paper, it is shown that every surjective isometry between the unit spheres of two finite dimensional C⁎C⁎-algebras extends to a real-linear Jordan ⁎-isomorphism followed by multiplication operator by a fixed unitary element. This gives an affirmative answer to Tingley's problem between two finite-dimensional C⁎C⁎-algebras. Moreover, we show that if two finite dimensional C⁎C⁎-algebras have isometric unit spheres, then they are ⁎-isomorphic.