Characterization Of Strong K-commutativity Preserving Maps

Let R be a ring(or an algebra over a field F). For a given positive integer k≥ 1, the k-commutator of A and B is defined by[A,B]k=[[A,B]k-1,B] with[A,B]0=A and [A,B]1=[A,B]=AB-BA.A map Φ:Râ†'R is said to be strong k-commutativity preserving if[Φ(A),Φ(B)]k=[A,B]k for all A,B∈R.Assume M2(F)be the algebra of 2×2 matrices over the real or complex field F. The main result in the second chapter of this paper is shown that a map Φ:M 2(F)â†'M 2(F) with range containing all rank one matrices is strong k-commutativity preserving if and only if there exist a functional h:M2(F)â†'F and a scalar λ∈F with λk+1=1 such that Φ(A)=λA+h(A)I for all A∈M2(F).LetX be a Banach space of dimension≥2 over the real or complex field F and A a standard operator algebra in B(X). The main result in the third chapter of this paper is shown that,if Φ is a surjective map on A,then Φ is strong 3-commutativity preserving if and only if there exist a functional h:Aâ†'F and a scalar λ∈F with λ4=1 such that Φ(A)=λA+h(A)I for all A∈4.