Fixed Point Property for Nonexpansive Mappings and Reflexivity: A Hilbert C ^∗ ‐Module Approach

We prove that for any infinite‐dimensional Hilbert space H , there exists a sequence of infinite‐dimensional, full Hilbert C ^∗ ‐modules over K ( H )—the C ^∗ ‐algebra of compact operators on H —each of which is isomorphic to H and has the fixed point property for nonexpansive mappings (FPP). In particular, within the class of Hilbert C ^∗ ‐modules this shows that there is no finite upper bound on the Banach–Mazur distance beyond which every full Hilbert C ^∗ ‐module isomorphic to a Hilbert space must fail to have the FPP. Moreover, we characterize reflexivity for full Hilbert C ^∗ ‐modules over commutative C ^∗ ‐algebras: such a module is a reflexive Banach space if and only if its underlying C ^∗ ‐algebra is finite‐dimensional. As a consequence, a full Hilbert C ^∗ ‐module over a commutative C ^∗ ‐algebra has the FPP precisely when it is reflexive as a Banach space.