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 del [x, y](n)= del[x, del[x, y](n-1)] with del[x, y](0) = y, del[x, y](1) = del[x, y] = xy- yx*, del[x, y](2) = x(2)y - 2xyx* + y(x*)2 and X lozenge(n) y = x lozenge (x lozenge(n-1) y) with x lozenge(0) y = y, x lozenge(1) y = x lozenge y = xy+ yx*, x lozenge(2) y = x(2)y + 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 del[x, T(x)](n) is an element of Z(R) (n = 1, 2) for all x is an element of R, then T(x) = lambda x* for all x is an element of R, where lambda is an element of C, the extended centroid of R. We also characterize Jordan left *-centralizers of prime rings with involution via skew Jordan product.