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.