Orthogonality preserving mappings on inner product C* -modules

Suppose that A is a C^*-algebra. We consider the class of A-linear mappins between two inner product A-modules such that for each two orthogonal vectors in the domain space their values are orthogonal in the target space. In this paper, we intend to determine A-linear mappings that preserve orthogonality. For this purpose, suppose that E and F are two inner product A-modules and A+ is the set of all positive elements of A. We show that an A-linear mapping T:E→F preserves orthogonality if and only if there exists a∈A+ such that ⟨Tx,Ty⟩= a^2 ⟨x,y⟩ for each x,y∈E. At first recall that two vector x,y∈E are ordinary orthogonal if ⟨x,y⟩=0 and then we introduce the notion of orthogonality in an inner product A-module in three ways and show that an A-linear mapping between two inner product A-modules preserves the ordinary orthogonality if and only if it preserves each one of the new orthogonality.