Approximately orthogonality preserving mappings on C∗-modules

We study orthogonality preserving and approximately orthogonality preserving mappings in the setting of inner product C∗-modules. In particular, if V and W are inner product C∗-modules over the C∗-algebra A, any scalar multiple of an A-linear isometry is an A-linear orthogonality preserving mapping. The converse does not hold in general, but it holds if A contains K(H) (the C∗-algebra of all compact operators on a Hilbert space H). Furthermore, we give the estimate of ‖〈Tx,Ty〉−‖T‖2〈x,y〉‖ for an A-linear approximately orthogonality preserving mapping T:V→W when V and W are inner product C∗-modules over a C∗-algebra containing K(H). In the case A=K(H) and V, W are Hilbert, we also prove that an A-linear approximately orthogonality preserving mapping can be approximated by an A-linear orthogonality preserving mapping.