Characterization Of Strong K-skew Commutativity Preserving Maps

Let R be a ring(or an algebra over a field IF)with an involution*.For a given positive integer K?1,the k-skew commutator of A and B is defined by*[A,B]k =*[A,*[A,B]k-1]1 with*[A,B]0 = B and*[A,B]1=[A,B]*= AB-BA*.A map ?:R?R is said to be strong k-skew commutativity preserving if*[?A),?(B)]k =*[A,B]k for all A,B ?R.The aim of the paper is to discuss the problem of how to characterize the strong k-skew commutativity preserving maps and the main results are as follows.(1)Let R be a unital prime*-ring containing a nontrivial symmetric idempotent,and?:R?R be a surjective map.We show that ? is strong 2(3)-skew commutativity preserving if and only if there exists ? E CS with ?3 = I(?4 = I)such that ?(A)= ?A for all A?TZ.Where I is the unit of R and CS is the symmetric extend centroid of R.For the case k= 3,some more mild condition assumed on R are needed.(2)Assume M2(F)be the algebra of 2×2 matrices over the real or complex field IF.A map ? over M2(F)with range containing all rank one projections is strong k-skew commutativity preserving if and only if there exists a scalar ??E IF with ?k+1 = 1 such that ?(A)= ?A for all A? M2(F).(3)Let H be a complex Hilbert space with dim H?3 and A be a self-adjoint standard operator algebra on H.Assume k?4 and ? is a map on A with range containing all rank one projections.Then ? is strong k-skew commutativity preserving if and only if there exists a scalar ? with?k+1such that ?(A)= ?A for all A?A.