Hilbert C$ ^* $-module Jordan Homomorphisms
Abstract
Let $ \mathcal{M} $ be a Hilbert C$ ^* $-module. A linear mapping $ \psi : \mathcal{M} \rightarrow \mathcal{M} $ is said to be a
Hilbert C$ ^* $-module Jordan homomorphism on $ \mathcal{M} $, if it satisfies the equation
$
\psi (\langle a,b \rangle a) =\langle \psi (a),\psi(b)\rangle \psi(a)
$
for all $ a,b \in \mathcal{M} $.
In this paper, we show that, if $ \mathcal{M} $ is prime, then every Hilbert C$ ^* $-module Jordan homomorphism $ \psi $ from $ \mathcal{M} $ onto $ \mathcal{M} $, is a Hilbert C$ ^* $-module homomorphism or a Hilbert C$ ^* $-module anti-homomorphism on $ \mathcal{M} $. Also we prove a similar result about generalized Hilbert C$ ^* $-module Jordan homomorphism.