Strong (skew) ξ-Lie commutativity preserving maps on algebras

In Bell and Daif (1994), the authors gave the conception of strong commutativity preserving maps. Let  be a subset of . A map Φ: →  is called strong commutativity preserving if [Φ(T),Φ(S)] = [T, S] for all T, S ∈ . Note that a strong commutativity preserving map must be commutativity preserving, but the inverse is not true generally. Bell and Daif (1994) proved that  must be commutative, if  is a prime ring and  admits a derivation or a non-identity endomorphism which is strong commutativity preserving on a right ideal of . Brešar and Miers (1994) proved that every strong commutativity preserving additive map Φ on a semiprime ring  is of the form Φ(A) = A + (A), where ∈ , the extended centroid of , 2 = 1, and : →  is an additive map. Recently, Lin and Liu (2008) obtained the similar result on a noncentral Lie ideal of a prime ring. Qi and Hou (2010; 2012) gave a complete characterization of strong commutativity preserving surjective maps (without the assumption of additivity) on prime rings and triangular algebras, respectively. *Corresponding author: Xiaofei Qi, Department of Mathematics, Shanxi University, Taiyuan 030006, P.R. China E-mail: xiaofeiqisxu@aliyun.com

In Bell and Daif (1994), the authors gave the conception of strong commutativity preserving maps.
Let  be a subset of . A map Φ: →  is called strong commutativity preserving if [Φ(T), Φ(S)] = [T, S] for all T, S ∈ . Note that a strong commutativity preserving map must be commutativity preserving, but the inverse is not true generally. Bell and Daif (1994) proved that  must be commutative, if  is a prime ring and  admits a derivation or a non-identity endomorphism which is strong commutativity preserving on a right ideal of . Brešar and Miers (1994) proved that every strong commutativity preserving additive map Φ on a semiprime ring  is of the form Φ(A) = A + (A), where ∈ , the extended centroid of , 2 = 1, and :  →  is an additive map. Recently, Lin and Liu (2008) obtained the similar result on a noncentral Lie ideal of a prime ring. Qi and Hou (2010; gave a complete characterization of strong commutativity preserving surjective maps (without the assumption of additivity) on prime rings and triangular algebras, respectively. Let  be a *-ring. For any A, B ∈ , [A, B] * = AB − BA * denotes the skew Lie product of A and B.
This kind of product is found playing a more and more important role in some research topics such as representing quadratic functionals with sesquilinear functionals, and its study has attracted many authors' attention (see [Brešar & Fosňer, 2000;Chebotar, Fong, & Lee, 2005;Cui & Hou, 2006] and the reference therein). Molnár (1996) initiated the systematic study of this skew Lie product, and studied the relation between subspaces and ideals of (H), the algebra of all bounded linear operators acting on a Hilbert space H.
Recall that a map Φ :  →  is called zero skew Lie product preserving, if Φ(A)Φ(B) − Φ(B)Φ(A) * =0 whenever AB − BA * = 0 for any A, B ∈ . Additive or linear maps preserving zero skew Lie products on various rings and algebras had been studied by many authors (see, Bell & Daif, 1994 and the references therein). More specially, Φ is strong skew commutativity preserving, if [Φ(A), Φ(B)] * = [A, B] * for all A, B ∈ . It is obvious that strong skew commutativity preserving maps must be zero skew Lie product preserving. However, the inverse is not true generally. In Cui and Park (2012), they proved that, if  is a factor von Neumann algebra, then every strong skew commutativity preserving map Φ on  has the form Φ Qi and Hou (2013) generalized the above result to von Neumann algebras without central summand of type I 1 .
Recall that A commutes with B up to a factor ∈ if AB = BA. Note that the concept of commutativity up to a factor for pairs of operators is important and has been studied in the context of operator algebras and quantum groups (see Brooke, Busch, &Pearson, 2002 andKassel, 1995).
Motivated by this, a binary operation [A, B] = AB − BA, called -Lie product of A and B, was introduced in Qi and Hou (2009). Thus, we also can define the skew -Lie product of A and B. Let  be a *-algebra over , where is a field with an involution * . For A, B ∈  and ∈ , we call AB − BA * the skew -Lie product of A and B. It is obvious that the skew -Lie product is the skew Lie product if = 1. Now, based on these concepts, we say that a map Φ : The purpose of this paper is to consider nonlinear strong (skew) -Lie commutativity preserving maps on general algebras with ≠ 1. Let  be any unital algebra over any field and ∈ with ≠ 1. Denote by () the center of . Assume that Φ :  →  is a map. In Section 2, we prove that Φ preserves strong -Lie commutativity if and only if Φ(I) ∈ (), Φ(I) 2 = I, and Φ(A) = Φ(I)A for all A ∈  (Theorem 2.1). In Section 3, we furthermore assume that  is a *-algebra. It is shown that, if Φ(I) = Φ(I) * , then Φ preserves strong skew -Lie commutativity if and only if Φ(I) ∈ (), Φ(I) 2 = I and Φ(A) = Φ(I)A for all A ∈  (Theorem 3.1); if | | = 1 and Φ is surjective, then Φ preserves strong skew -Lie commutativity if and only if Φ(I) = Φ(I) * ∈ (), Φ(I) 2 = I and Φ(A) = Φ(I)A for all A ∈  (Theorem 3.2).

Maps preserving strong -Lie commutativity
In this section, we will give a characterization of nonlinear strong -Lie commutativity preserving maps on general algebras. The following is our main result.
Theorem 2.1 Let  be any algebra with unit I over a field , and let ∈ with ≠ 1. Assume that Φ: →  is a map. Then Φ preserves strong -Lie commutativity, that is, In the sequel, we will complete the proof by considering two cases.
Case 1 = −1. Combining Case 1 and Case 2, the proof of the theorem is complete.

Maps preserving strong skew -Lie commutativity
In this section, we will discuss the maps preserving strong skew -Lie commutativity on general algebras.
Theorem 3.1 Let  be any *-algebra with unit I over the real or complex field and let ∈ with ≠ 1. Assume that Φ :  →  is a map. If Φ(I) = Φ(I) * , then Φ preserves strong skew -Lie commutativity, that is, Φ satisfies Φ In the following, we will prove the theorem by two cases.
Case 1 | | ≠ 1.  We complete the proof of the theorem.
Theorem 3.2 Let  be any *-algebra with unit I over the real or complex field , and let ∈ with ≠ 1 and | | = 1. Assume that Φ: →  is a surjective map. Then Φ preserves strong skew -Lie commutativity if and only if Φ(I) = Φ(I) * ∈ (), Φ(I) 2 = I and Φ(A) = Φ(I)A for all A ∈ .