Approximately orthogonality preserving mappings on [formula omitted]-modules

Abstract

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 , T y 〉 − ‖ 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.