Characterization of additive *-homomorphisms and Jordan *-homomorphisms on operator ideals

Abstract.

Let H be a separable complex Hilbert space and let $\cal {B (H)}$ denote the algebra of all bounded linear operators on H. It is proved that if $\cal {I} \subset \cal {B (H)}$ is an ideal, then the additive function $\Phi : \cal {I} \to \cal {B (H)}$ is a Jordan *-homomorphism if and only if it satisfies¶¶$ \Phi (A^*A + AA^*) = \Phi (A)^* \Phi (A) + \Phi (A) \Phi(A)^* \qquad (A \in \cal {I}). $¶Similarly, $\Phi$ is a *-homomorphism if and only if it satisfies¶¶$ \Phi (A^*A) = \Phi (A)^* \Phi (A) \qquad (A \in \cal {I}). $¶These functional equations come from noncommutative approximation theory.