Communications of the
Korean Mathematical Society CKMS

We show that mappings preserving unit distance are close to two-isometries. We also prove that a mapping $f$ is a linear isometry up to translation when $f$ is a two-expansive surjective mapping preserving unit distance. Then we apply these results to consider two-isometries between normed spaces, strictly convex normed spaces and unital $C^*$-algebras. Finally, we propose some remarks and problems about generalized two-isometries on Banach spaces.

Keywords: Alesandrov problem, $C^*$-algebra, Mazur-Ulam theorem, strictly convex, two-isometry, two-expansive mapping

MSC numbers: 46B04, 47A62, 47B99, 52A07