On Jordan left-I-centralizers of prime and semiprime gamma rings with involution

Let M   be a 2-torsion free ΓΓ-ring with involution I   satisfying the condition xαyβz=xβyαzxαyβz=xβyαz for all x,y,z∈Mx,y,z∈M and α,β∈Γα,β∈Γ. The object of our paper is to show that every Jordan left-I  -centralizer on a semiprime ΓΓ-ring with involution I, is a reverse left-I  -centralizer. We solve some functional equations in prime and semiprime ΓΓ-rings with involution I by means of the above results. Moreover, we discuss some more related results.