Jordan left *-centralizers of prime and semiprime rings with involution

Abstract

Let R be a ring with involution ′*′. An additive mapping T : R → R is called a left *-centralizer (resp. Jordan left *-centralizer) if T(xy) = T(x)y* (resp. T(x 2) = T(x)x*) holds for all $${x,y \in R}$$ , and a reverse left *-centralizer if T(xy) = T(y)x* holds for all $${x,y\in R}$$ . In the present paper, it is shown that every Jordan left *-centralizer on a semiprime ring with involution, of characteristic different from two is a reverse left *-centralizer. This result makes it possible to solve some functional equations in prime and semiprime rings with involution. Moreover, some more related results have also been discussed.