A fixed point approach to the stability of $\varphi$-morphisms on Hilbert $C^*$-modules

Let E, F be two Hilbert C∗-modules over C∗-algebras A and B respectively. In this paper, by the alternative fixed point theorem, we give the Hyers-Ulam-Rassias stability of the equation 〈U(x), U(y)〉 = φ(〈x, y〉) (x, y ∈ E), where U : E → F is a mapping and φ : A → B is an additive map. 1 School of Mathematics and Computer Sciences, Damghan University, Damghan, Iran. E-mail address: abbaspour@du.ac.ir 2 School of Mathematics and Computer Sciences, Damghan University, Damghan, Iran. E-mail address: md ramezanpour@yahoo.com, ramezanpour@du.ac.ir Date: Received: 25 July 2010; Revised: 22 September 2010; Accepted: 29 September 2010. ∗ Corresponding author. 2010 Mathematics Subject Classification. Primary 39B82; Secondary 46L08.