Special symmetries of Banach spaces isomorphic to Hilbert spaces
Volume 200 / 2010
Studia Mathematica 200 (2010), 31-40 MSC: Primary 46C15, 43A35; Secondary 46B08, 43A07. DOI: 10.4064/sm200-1-2
We characterize Hilbert spaces among Banach spaces in terms of transitivity with respect to nicely behaved subgroups of the isometry group. For example, the following result is typical: If X is a real Banach space isomorphic to a Hilbert space and convex-transitive with respect to the isometric finite-dimensional perturbations of the identity, then X is already isometric to a Hilbert space.