Results on _α_centralizers of prime and semiprime rings with involution

Let R be a prime or semiprime ring equipped with an involution ∗ and α be an automorphism of R. An additive mapping T : R → R is called a left (resp. right) α−∗centralizer of R if T (xy) = T (x)α (y∗) (resp. T (xy) = α (x∗)T (y)) holds for all x, y ∈ R, where α is an endomorphism of R. A left (resp. right) Jordan α−∗centralizer T : R → R is an additive mapping such that T ( x2 ) = T (x)α(x∗) (resp. T ( x2 ) = α(x∗)T (x)) holds for all x ∈ R. In this paper, we obtain some results about Jordan α−∗centralizer of R with involution.