A note on semiderivations in prime rings and C*-algebras

Let R be a prime ring with the extended centroid C and the Matrindale quotient ring Q. An additive mapping F:R→R is called a semiderivation associated with a mapping G:R→R, whenever F(xy)=F(x)G(y)+xF(y)=F(x)y+G(x)F(y) and F(G(x))=G(F(x)) holds for all x,y∈R. In this manuscript, we investigate and describe the structure of a prime ring R which satisfies F(xm∘yn)∈Z(R) for all x,y∈R, where m,n∈Z+ and F:R→R is a semiderivation with an automorphism ξ of R. Further, as an application of our ring theoretic results, we discussed the nature of C∗-algebras. To be more specific, we obtain for any primitive C∗-algebra A. If an anti-automorphism ζ:A→A satisfies the relation (xn)ζ+xn∗∈Z(A) for every x,y∈A, then A is C∗-W4-algebra, i.e., A satisfies the standard identity W4(a1,a2,a3,a4)=0 for all a1,a2,a3,a4∈A.