We give a sufficient condition for a Banach space with which the homogeneous extension of a surjective isometry from the unit sphere of it onto another one is real-linear. The condition is satisfied by a uniform algebra and a certain extremely $C$-regular space of real-valued continuous functions.
Mankiewicz's theorem and the Mazur--Ulam property for C*-algebras
A Note on The Mazur-Ulam Property of Almost-CL-spaces
Every 2-dimensional Banach space has the Mazur-Ulam property
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