On the extension of isometries between unit spheres of \(E\) and \(C(\Omega)\). (Q1421086)

The main result in this paper asserts that a surjective isometry from the unit sphere of a Banach space, whose smooth points are dense, on the unit sphere of the space of continuous functions on a compact Hausdorff space, satisfying a certain condition (which is necessary), can be extended to a linear isometry between the corresponding spaces. The author obtains several corollaries by proving that the class of Banach spaces whose set of smooth points is dense includes \( L_{1}({\mu})\) (\({\mu}\) \({\sigma}\)-finite), \( {\mathcal C}(K)\) (\(K\) compact Hausdorff) and weakly compactly generated spaces (in particular, separable and reflexive spaces).