Jordan homomorphisms on Hilbert C∗-modules

Abstract

We generalize the concept of homomorphisms between Hilbert C${}^{\ast }$-modules to the concept of Jordan homomorphisms. Let $M$ be a Hilbert C${}^{\ast }$-module over a C${}^{\ast }$-algebra $A$ and $\phi :M\to M$ be a map. Under the condition that $A$ is commutative and $\phi$ is $ℂ$-linear bounded, we show that $\phi$ is a Jordan homomorphism if and only if $\phi$ is a homomorphism. In addition, we also discuss the relationship between $\Phi$-unitary maps and automorphisms, and give some conditions under which $\phi$ is an automorphism if and only if $\phi$ is a $\Phi$-unitary map.