Descripción
A Banach space X satisfies the Mazur–Ulam property if for any Banach space Y , every surjective isometry ∆ : S(X) → S(Y ) admits an extension to a surjective real linear isometry from X onto Y , where S(X) and S(Y ) denote the unit spheres of X and Y , respectively. An equivalent reformulation tells that X satisfies the Mazur–Ulam property if the so-called Tingley’s problem admits a positive solution for every surjective isometry from S(X) onto the unit sphere of any Banach space Y . We shall make in this talk a brief incursion into the origin of the quoted extension problems and provide a recent state of the art of these questions. We shall also present some of the strategies normally used for solving them by studying the Mazur–Ulam property in the C∗-algebra C(K) of all continuous complex-valued functions defined on a Stonean space K.