A characterization of jordan left ∗ -centralizers via skew lie and jordan products

Let n≥ 1 be a fixed integer and R be a ring with involution ‘∗ ’. For any two elements x and y in R, the n-skew Lie product and n-skew Jordan product are given by ▿ [x, y] n= ▿ [x, ▿ [x, y] n-1] with ▿[x,y]0=y,▿[x,y]1=▿[x,y]=xy-yx∗,▿[x,y]2=x2y-2xyx∗+y(x∗)2 and x⋄ ny= x⋄ (x⋄ n-1y) with x⋄ y= y, x⋄ 1y= x⋄ y= xy+ yx∗, x⋄2y=x2y+2xyx∗+y(x∗)2. The purpose of this paper is to characterize Jordan left ∗ -centralizers satisfying certain functional identities involving skew Lie product and skew Jordan products. In particular, it is proved that if R is a 2-torsion free prime ring with involution of the second kind admits a non-zero Jordan left ∗ -centralizer T such that ▿ [x, T(x)] n∈ Z(R) (n= 1 , 2) for all x∈ R, then T(x) = λx∗ for all x∈ R, where λ∈ C, the extended centroid of R. We also characterize Jordan left ∗ -centralizers of prime rings with involution via skew Jordan product.