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 T x,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. © 2007 Elsevier Inc. All rights reserved.