A Characterization of Jordan Left $^*$-Centralizers Via Skew Lie and Jordan Products

Abstract

Let $n\ge 1$ 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 $\triangledown [x,y]_n=\triangledown [x,\triangledown [x,y]_{n-1}]$ with $\triangledown [x,y]_0=y, ~\triangledown [x,y]_1= \triangledown [x,y]=xy-yx^* ,~ \triangledown [x,y]_2=x^2y-2xyx^*+y(x^*)^2$ and $x\diamond _ny=x\diamond (x\diamond _{n-1}y)$ with $x\diamond _0y=y$, $x\diamond _1y=x\diamond y=xy+yx^*$, $x\diamond _2 y=x^2y+2xyx^*+y(x^*)^2$. 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 $\triangledown [x,T(x)]_n\in Z(R)(n = 1, 2)$ for all $x\in R$, then $T(x)=\lambda x^*$ for all $x\in R$, where $\lambda \in C$, the extended centroid of R. We also characterize Jordan left $*$-centralizers of prime rings with involution via skew Jordan product.