On isometries between unit spheres of abstract \(M\) spaces (Q2725224)

D. Tingley raised the question whether any isometric map from the unit sphere \(S_{X}\) of a normed space \(X\) to the unit sphere \(S_{Y}\) of a normed space \(Y\) is the restriction of a linear map from \(X\) to \(Y\). This paper gives an affirmative answer to Tingley's problem for the case of abstract \(M\) spaces, which are by definition, Banach lattices \(E\) such that \(\|x\vee y\|=\max \{ \|x\|,\|y\|\} \) for any \(x,y\geq 0\) in \(E\). More precisely, the author shows that any isometric map from the unit sphere \( S_{X}\) of an abstract \(M\) space \(X\) onto the unit sphere \(S_{Y}\) of an abstract \(M\) space \(Y\) can be extended to an \(\mathbb{R}\)-linear isometry from \( X\) onto \(Y\).