Strong Biskew Commutativity Preserving Maps On Operator Algebras

General preserving problem on operator algebra are one of the most active fields of operator theory and operator algebra,and have received many attention.Let A be an algebra with an involution*.For a given positive integer k≥1,the k-biskew Lie product or 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*.Let Φ:A→A be a map.Then Φ is said to be strong k-biskew commutativity preserving if ◇[Φ(A),Φ(B)]k=◇[A,B]k for all A,B∈A.In particular,when k=1,the 1-biskew Lie product is called bi-skew Lie product,and the corresponding map Φ is called strong bi-skew commutativity preserving.The aim of the thesis is to discuss the problem of how to characterize the strong k-biskew(k=1,2)commutativity preserving maps on operator algebras.The main results are as follows.(1)Let A be a prime*-algebra over a field IF containing the unit I and a nontrivial symmetric idempotent.Assume that the involution*is of the second kind and Φ:A→A is a surjective map.If Φ(I)is unitary,then Φis strong bi-skew commutativity preserving if and only if Φ(A)=αAΦ(I)holds for all A ∈ A,where α is an element in the symmetric extended centroid of A with α2=1;If the characteristic of the field IF is not 2 and Φ(I)is a self-adjoint central element in A,then Φ is strong 2-biskew commutativity preserving if and only if Φ(I)3=I and Φ(A)=Φ(I)A for all A E A.Based on these,complete characterizations of strong bi-skew commutativity and 2-biskew commutativity preserving surjective maps on factor von Neumann algebras are respectively obtained.(2)Let M be a von Neumann algebra without central summands of type I1.Assume that Φ:M→M is a surjective map.If Φ(I)is unitary,then Φ is strong bi-skew commutativity preserving if and only if there exists a self-adjoint central element Z ∈ M with Z2=I such thatΦ(A)=ZAΦ(I)for all A∈M.