The solution of Tingley's problem for the operator norm unit sphere of complex n × n matrices

In this paper, we study geometric structures and isometries of the unit sphere of n×n complex matrices. We introduce the notion of proper exposed points of the unit ball of a Banach space which provides a refinement of that of exposed point provided that the space is separable. As a new geometric interpretation of unitary matrices, it is shown that the set of all proper exposed points of the unit ball of Mn(C) coincides with its unitary group U(n), where Mn(C) denotes the algebra of all n×n complex matrices. Moreover, we show that every maximal convex subset of the unit sphere of Mn(C) can be identified with the unit ball of Mn−1(C) by an affine bijection preserving the unitary matrices. This map is isometric with respect to any unitarily invariant norm. As an application, we completely determine the forms of surjective isometries on the unit sphere of Mn(C) with respect to the operator norm, which provides an affirmative answer to Tingley's problem on Mn(C).