Hilbert C∗-Module Jordan Homomorphisms

Let ℳ be a Hilbert C^∗-module. A linear mapping ψ : ℳ→ ℳ is said to be a Hilbert C^∗-module Jordan homomorphism on ℳ if it satisfies the equation ψ(〈a, b〉a)= 〈ψ(a),ψ(b)〉 ψ(a) for all a, b ∈ℳ. In thispaper,we show that if ℳ is prime, then every Hilbert C^∗-module Jordan homomorphism ψ from ℳ onto ℳ is a Hilbert C^∗-module homomorphism or a Hilbert C^∗-module anti-homomorphism on ℳ. We also prove a similar result about generalized Hilbert C^∗-module Jordan homomorphism.