The celebrated Mazur-Ulam theorem states that a surjective isometry between two Banach space is a real-linear map followed by a translation. Here isometry means just a distance preserving map, without assuming any algebraic property. Mankiewicz localized this theorem in the sense that a surjective isometry between connected open sets of Banach spaces is extended to a surjective affine isometry between these Banach spaces. He also proved that the same holds for a surjective isometry between convex bodies. These theorems inspired Tingley to ask whether a surjective isometry between the unit spheres of Banach spaces is extended to a surjective real-linear isometry between whole spaces. Although I do not exhibit each of literatures, a considerable number of interesting results have shown that Tingley's problem has affirmative answers. No counterexample is known. Following Cheng and Dong we say that a real Banach space $E$ has the Mazur-Ulam property if a surjective isometry between the unit spheres of $E$ and that of any real Banach space is extended to a surjective real-linear isometry between whole spaces. The Mazur-Ulam property is nowadays a challenging subject of study. In this talk we ask if a uniform algebra satisfies the Mazur-Ulam property. Oi, Shindo Togashi and myself already proved that a surjective isometry between uniform algebas is extended to a surjective real-linear isometry between these uniform algebras. A complex Banach space $B$ has the {\it complex} Mazur-Ulam property if a surjective isometry between the unit spheres of $B$ and that of any {\it complex} Banach space is extended to a surjective real-linear isometry between whole spaces. If a complex Banach space $B$, as a real Banach space, has the Mazur-Ulam property, then $B$ has the complex Mazur-Ulam propety. We do not know if the converse holds or not. We show that a uniform algebra has the complex Mazur-Ulam property.