Strong 2-skew Commutativity Preserving Maps on Prime Rings with Involution

Abstract

Let ${\mathcal {R}}$ be a unital prime $*$-ring containing a nontrivial symmetric idempotent. For $A,B\in {\mathcal {R}}$, the skew commutator and 2-skew commutator are defined, respectively, by ${}_*[A,B]=AB-BA^*$ and ${}_*[A,B]_2= {{}_*[A, {{}_*[A,B]}]}$. Let $\Phi :{\mathcal {R}} \rightarrow {\mathcal {R}}$ be a surjective map. We show that (1) $\Phi$ satisfies ${}_*[\Phi (A),\Phi (B)] = {{}_*[A,B] }$ for all $A, B\in {\mathcal {R}}$ if and only if there exists $\lambda \in \{-1,1\}$ such that $\Phi (A)=\lambda A$ for all $A\in {\mathcal {R}}$; (2) $\Phi$ satisfies ${}_*[\Phi (A),\Phi (B)]_2= {{}_*[A,B]_2}$ for all $A, B\in {\mathcal {R}}$ if and only if there exists $\lambda \in {\mathcal {C}}_S$ with $\lambda ^{3} = I$ such that $\Phi (A) = \lambda A$ for all $A \in {\mathcal {R}}$, where I is the unit of ${\mathcal {R}}$ and ${\mathcal {C}}_S$ is the symmetric extend centroid of ${\mathcal {R}}$. This is then applied to prime $\hbox {C}^*$-algebras, factor von Neumann algebras and indefinite self-adjoint standard operator algebras to get a complete invariant for the identity map and to symmetric standard operator algebras as well as matrix algebras.

1 Introduction

The main purpose of the present paper is to discuss the problem of characterizing the strong skew commutativity preserving maps and the strong 2-skew commutativity preserving maps on operator algebras. However, we discuss first in a general pure algebraic frame and then apply to operator algebras.

Let ${\mathcal {R}}$ be a ring. For any elements, $A,B\in {\mathcal {R}}$, define $[A,B]_0=A, [A,B]_1=AB-BA$, and inductively $[A,B]_k=[[A,B]_{k-1},B]$, where $k\ge 1$ is a positive integer. A map $\Phi :{\mathcal {R}} \rightarrow {\mathcal {R}}$ is said to be strong k-commutativity preserving if $[\Phi (A),\Phi (B)]_k = [A,B]_k$ for all $A,B\in {\mathcal {R}}$. The problem of characterizing strong k-commutativity preserving maps has been studied intensively on various rings and operator algebras (for example, ref. [1,2,3,4,5,6,7,8]). It was shown in [5] (resp. [6]) that a surjective map on a unital prime ring ${\mathcal {R}}$ containing a nontrivial idempotent is a strong commutativity preserving map (resp. a strong 2-commutativity preserving map) if and only if it has the form: $A\mapsto \lambda A+f(A)$, where $\lambda \in {\mathcal {C}}$ with $\lambda ^2=1$ (resp. $\lambda ^3=1$), $f:{\mathcal {R}}\rightarrow {{\mathcal {Z}}}({\mathcal {R}})$ is a map, ${\mathcal {C}}$ is the extended centroid of ${\mathcal {R}}$, and ${{\mathcal {Z}}}({\mathcal {R}})$ is the center of ${\mathcal {R}}$.

Recall that a ring ${\mathcal {R}}$ is called a $*$-ring or a ring with involution if there is an additive map $*: \mathcal {R}\rightarrow {\mathcal {R}}$ satisfying $(AB)^{*}=B^{*}A^{*}$ and $(A^{*})^{*}=A$ for all $A, B\in {\mathcal {R}}$. For $A,B\in {\mathcal {R}}$, the skew Lie product or the skew commutator of A and B is denoted by ${}_*[A,B]=AB-BA^{*}$ (Ref. [9], where ${}_*[A,B]$ is denoted by $[A,B]_*$). The concept of skew Lie product is found playing an important role in some research topics, and their study has attracted many authors’ attention (Ref. [10,11,12,13,14,15,16,17,18]). Similar to the commutator case stated in previous paragraph, a map $\Phi :{{\mathcal {R}}} \rightarrow {{\mathcal {R}}}$ is said to be strong skew commutativity preserving if ${}_*[\Phi (A), \Phi (B)]$ = ${}_*[A, B]$ for any $A, B\in {{\mathcal {R}}}$. For the problem of how to characterize the strong skew commutativity preserving maps, it was shown in [12] that a nonlinear surjective map $\Phi$ on a factor von Neumann algebra ${\mathcal {A}}$ is strong skew commutativity preserving if and only if it has the form $A\mapsto \Psi (A)+h(A)I$, where $\Psi$ is a strong skew commutativity preserving bijective linear map and $h:\mathcal A\rightarrow \mathbb {R}$ is a functional with $h(0) = 0$. As usual, $\mathbb R$ and ${\mathbb {C}}$ denote the real number field and the complex number field, respectively. In [16], a more precise form of the above surjective map was achieved: $A\mapsto ZA$, where $Z\in {\mathcal {Z}}_S$, the set of all self-adjoint elements in the center of ${\mathcal {A}}$. In a pure algebraic frame, it was also shown in [16] that if ${\mathcal {R}}$ is a prime $*$-ring which contains a nontrivial symmetric idempotent, then the strong skew commutativity preserving surjective maps on ${\mathcal {R}}$ have the form $\Phi (A)=ZA+f(A)$ for all $A\in {{\mathcal {R}}}$, where $Z\in {\mathcal {Z}}_S({\mathcal {R}})$ and $f:{\mathcal {R}}\rightarrow \mathcal Z_S({\mathcal {R}})$. Recently, [17] and [18] generalized this result to unital $*$-rings ${\mathcal {R}}$ which contain a nontrivial symmetric idempotent P satisfying $A\mathcal {R}P=\{0\}\Rightarrow A=0$ and $A{\mathcal {R}}(I-P ) =\{ 0\}\Rightarrow A = 0$.

Let ${\mathcal {R}}$ be a $*$-ring. For any positive integer $k\ge 1$, like the concept of k-commutator, the k-skew commutator of a pair A, B in ${\mathcal {R}}$ is naturally defined inductively by ${}_*[A,B]_{k}={{}_*[A,{{}_*[A,B]_{k-1}}]}$ with ${}_*[A,B]_{0}=B,\,{{}_*[A,B]_{1}}={{}_*[A,B]}=AB-BA^*$. The k-skew commutator has some interesting properties. For instance, if ${\mathcal {R}}$ is unital, prime and of characteristic not 2, then ${}_*[A,B]_2=0$ holds for all B if and only if $A\in {\mathcal Z}={{\mathcal {Z}}}({\mathcal {R}})$ and $A=A^*$, that is, $A\in {\mathcal Z_S}$ (Lemma 4.3); if, in addition, ${\mathcal {R}}$ contains a nontrivial symmetric idempotent, then ${}_*[B,A]_2=0$ holds for all B if and only if $A=0$ (Lemma 4.2).

Let ${\mathcal {R}}$ be a $*$-ring. Recall that a map $\Phi :{{\mathcal {R}}} \rightarrow {{\mathcal {R}}}$ is said to be strong k-skew commutativity preserving if ${}_*[\Phi (A), \Phi (B)]_k={{}_*[A, B]_k}$ holds for all $A, B\in {{\mathcal {R}}}$. That is, the strong k-skew commutativity preserving maps are those maps that preserve the k-skew commutators. It is interesting to ask how to characterize the strong k-skew commutativity preserving maps. In the present paper, we focus our attention on the case $1\le k\le 2$.

Our aim is twofold. The first one is to sharp the results in [16,17,18] by proving the symmetric center valued map f is 0, and then get a characterization of the strong skew commutativity preserving maps. Let ${\mathcal {R}}$ be a unital $*$-ring which contains a nontrivial symmetric idempotent P satisfying $A{\mathcal {R}}P=0\Rightarrow A=0$ and $A{\mathcal {R}}(I-P ) = 0\Rightarrow A = 0$, where I denotes the unit. We show that a surjective map $\Phi :{{\mathcal {R}}}\rightarrow {{\mathcal {R}}}$ is strong skew commutativity preserving if and only if there exists $Z \in {{\mathcal {Z}}}_S$ with $Z^2=I$ such that $\Phi (A)=ZA$ for all $A\in {\mathcal {R}}$ (Theorem 2.1). Particularly, if ${\mathcal {R}}$ is prime, we have further that $Z\in \{-I,I\}$. The second one, as the main part of the paper, is to characterize the strong 2-skew commutativity preserving maps. Let ${\mathcal {R}}$ be a unital prime $*$-ring with the unit I and containing a nontrivial symmetric idempotent. We show that a surjective map $\Phi :\mathcal {R}\rightarrow {\mathcal {R}}$ is strong 2-skew commutativity preserving if and only if there exists an element $\lambda \in {\mathcal {C}}_{S}$ with $\lambda ^{3}=I$ such that $\Phi (A)=\lambda A$ for all $A\in {{\mathcal {R}}}$. Applying to some operator algebras such as prime $\hbox {C}^*$-algebras, factor von Neumann algebras and indefinite self-adjoint standard operator algebras, we get a characterization of the identity map. It is somewhat surprising by our results that the 2-skew commutator is a complete invariant for the identity map for the operator algebras mentioned above. A characterization of strong 2-skew commutativity preserving maps on symmetric standard operator algebras is also established.

The paper is organized as follows. Section 2 is devoted to sharping the results in [16,17,18] and characterizing the strong skew commutativity preserving maps. In Sect. 3, we give a characterization of strong 2-skew commutativity preserving maps on unital prime $*$-rings containing a nontrivial symmetric idempotent (Theorem 3.1) and its corollaries applying to some operator algebras. Section 4 is devoted to proving Theorem 3.1 for the case when the ring is 2-torsion-free; while Sect. 5 is devoted to proving Theorem 3.1 for the case when the characteristic of the ring is 2.

2 Strong Skew Commutativity Preserving Maps

In this section, we sharp the main results in [16,17,18] by proving the symmetric center valued map f vanishes and consequently get a characterization of strong skew commutativity preserving maps.

Recall that a nontrivial symmetric idempotent P is an element so that $P^2=P=P^*$ and $P\not \in \{0,I\}$, where I is the unit.

Theorem 2.1

Let ${\mathcal {R}}$ be a unital ring with involution $*$. Assume that ${\mathcal {R}}$ contains a nontrivial symmetric idempotent P which satisfies:

• $(\mathbb {Q}_1)\ A{\mathcal {R}}P = \{0\}$ implies $A = 0, (\mathbb {Q}_2)\ A{\mathcal {R}}(I-P) =\{ 0\}$ implies $A = 0$.

Let $\Phi :{\mathcal {R}}\rightarrow {\mathcal {R}}$ be a surjective map. Then $\Phi$ is strong skew commutativity preserving if and only if there exists an element $Z \in {\mathcal {Z}}_{S }(\mathcal {R})$ with $Z^2 = I$ such that $\Phi (A)=ZA$ for all $A\in {\mathcal R}$.

Proof

The “if” part is obvious. To check the “only if” part, assume that $\Phi$ is strong skew commutativity preserving. By [17, 18], there exists a map $f:{\mathcal {R}}\rightarrow \mathcal {Z}_S({\mathcal {R}})$ and an element $Z\in {\mathcal {Z}}_S(\mathcal {R})$ with $Z^2=I$ such that $\Phi (A)=ZA+f(A)$ for all $A\in \mathcal {R}$. Now we need to show $f(A)=0$ for all $A\in {\mathcal {R}}$. Assume, on the contrary, there exists $B\in {\mathcal {R}}$ such that $f(B)\not =0$. For any $A\in {\mathcal {R}}$, as $\Phi$ is strong skew commutativity preserving, we have

$$\begin{aligned} {}_*[A,B]= & {} {{}_*[\Phi (A),\Phi (B)]}={{}_*[ZA+f(A),ZB+f(B)]}\\= & {} {{}_*[ZA,ZB]}+{{}_*[ZA,f(B)]}, \end{aligned}$$

which implies

$$\begin{aligned} Zf(B)(A-A^*)=(A-A^*)Zf(B)=0 \end{aligned}$$

for all $A\in {\mathcal {R}}$. Since $Z^2=I$,

$$\begin{aligned} f(B)(A-A^*)=(A-A^*)f(B)=0 \end{aligned}$$

(2.1)

for all $A\in {\mathcal {R}}$.

Obviously, Eq. (2.1) implies that

$$\begin{aligned} f(B)PA(I-P)+f(B)(I-P)A^*P=0 \end{aligned}$$

holds for every $A\in {{\mathcal {R}}}$. By the Peirce decomposition relative to P, this implies $f(B)PA(I-P)=0$ for all $A\in {\mathcal R}$. Hence, by (${{\mathbb {Q}}}_2$), we have $f(B)P=0$. Since $f(B)\in {{\mathcal {Z}}}({{\mathcal {R}}})$, this entails $f(B)AP=Af(B)P=0$ for all $A\in {{\mathcal {R}}}$, so that, by (${{\mathbb {Q}}}_1$), $f(B)=0$, a contradiction. Hence $f=0$ and $\Phi (A)=ZA$ for all A. $\square$

In the case when the ring is prime, a more concrete form is achieved.

Corollary 2.2

Let ${\mathcal {R}}$ be a unital prime ring with involution $*$ and containing a nontrivial symmetric idempotent. Let $\Phi :{\mathcal {R}}\rightarrow {\mathcal {R}}$ be a surjective map. Then $\Phi$ is strong skew commutativity preserving if and only if either $\Phi (A)= A$ for all $A\in {{\mathcal {R}}}$, or $\Phi (A)= - A$ for all $A\in {{\mathcal {R}}}$.

Proof

By Theorem 2.1, there exists a central symmetric element Z with $Z^2=I$ such that $\Phi (A)=ZA$ for each $A\in {\mathcal {R}}$. If $Z\not \in \{-I,I\}$, then $I+Z\not =0, I-Z\not =0$ and $(I+Z)(I-Z)=0$. It follows that $(I+Z)A(I-Z)=(I+Z)(I-Z)A=0$ holds for all $A\in {\mathcal {R}}$, which contradicts the primeness of ${\mathcal {R}}$. $\square$

Corollary 2.2 can be applied to prime $\hbox {C}^*$-algebras, factor von Neumann algebras as well as indefinite self-adjoint standard operator algebras like what we have done in the next section for characterizing strong 2-skew commutativity preserving maps (see Corollaries 3.2–3.7). We leave these to the readers.

3 Strong 2-skew Commutativity Preserving Maps

Begining from this section, we discuss the problem of how to characterize strong 2-skew commutativity preserving maps on prime $*$-rings and operator algebras.

Let ${\mathcal {R}}$ be a ring. Denote by ${\mathcal {Q}}=\mathcal {Q}_\mathrm{ml}({\mathcal {R}})$ the maximal left ring of quotients. The center ${\mathcal {C}}={\mathcal {C}}({\mathcal {R}})={\mathcal Z}({\mathcal {Q}})$ of ${\mathcal {Q}}$ is called the extended centroid of ${\mathcal {R}}$. Note that if ${\mathcal {R}}$ is prime, then ${\mathcal {Q}}$ is also prime and ${\mathcal {C}}$ is a field. If ${\mathcal {R}}$ is a prime $*$-ring, the symmetric extended centroid ${\mathcal {C}}_S=\{\lambda \in {\mathcal {C}}:\lambda =\lambda ^*\}$ is also a field. Moreover, ${\mathcal {Z}}({\mathcal {R}})\subseteq \mathcal {C}$ (Ref. [22] for details). Bear in mind that

$$\begin{aligned} {}_*[A,B]_2=A^2B-2ABA^*+B(A^*)^2. \end{aligned}$$

The following is our main result.

Theorem 3.1

Let ${\mathcal {R}}$ be a unital prime ring with involution $*$ and containing a nontrivial symmetric idempotent. Let $\Phi :{\mathcal {R}}\rightarrow {\mathcal {R}}$ be a surjective map. Then $\Phi$ is strong 2-skew commutativity preserving if and only if there exists $\lambda \in {{\mathcal {C}}}_S$ with $\lambda ^3=I$ such that $\Phi (A)=\lambda A$ for all $A\in {{\mathcal {R}}}$.

Denote by char$({\mathcal {R}})$ the characteristic of the ring ${\mathcal {R}}$. The proof of Theorem 3.1 will be given in Sect. 4 for the case char$({\mathcal {R}})\not =2$ and in Sect. 5 for the case char$({{\mathcal {R}}})=2$.

Now we give some applications of Theorem 3.1 to operator algebras.

By [19, Corollary 2.4], if ${\mathcal {A}}$ is a prime $\hbox {C}^*$-algebra, then its extended centroid ${{\mathcal {C}}}({\mathcal A})={\mathbb {C}}$. Thus, ${{\mathcal {C}}}_S({\mathcal {A}})={{\mathbb {R}}}$ and the following corollary is immediate, which also gives a characterization of the identity map on such prime $\hbox {C}^*$-algebras.

Corollary 3.2

Let ${\mathcal {A}}$ be a unital prime $\hbox {C}^*$-algebra containing a nontrivial self-adjoint idempotent and $\Phi :{{\mathcal {A}}}\rightarrow {\mathcal {A}}$ a surjective map. Then $\Phi$ is strong 2-skew commutativity preserving if and only if $\Phi$ is the identity map.

Recall that a von Neumann algebra ${{\mathcal {M}}}$ is a $\hbox {C}^*$-subalgebra of some ${{\mathcal {B}}}(H)$, the algebra of all bounded linear operators acting on a complex Hilbert space H, which satisfies the double commutant property: ${\mathcal M}^{\prime \prime }={\mathcal {M}}$, where ${{\mathcal {M}}}^\prime =\{T: T\in {{\mathcal {B}}}(H)\ \text{ and }\ TA=AT\ \forall A\in {{\mathcal {M}}}\}$ and ${{\mathcal {M}}}^{\prime \prime }=\{{{\mathcal {M}}}^\prime \}^\prime$. ${{\mathcal {M}}}$ is called a factor if its center ${\mathcal Z}({{\mathcal {M}}})={{\mathcal {M}}}\cap {{\mathcal {M}}}^\prime ={{\mathbb {C}}}I$. It is well known that von Neumann algebras are unital $\hbox {C}^*$-algebras containing many nontrivial self-adjoint idempotents, and every factor von Neumann algebra is prime. So the following corollary is obvious from Corollary 3.2.

Corollary 3.3

Let ${{\mathcal {M}}}$ be a factor von Neumann algebra and $\Phi : {{\mathcal {M}}}\rightarrow {{\mathcal {M}}}$ be a surjective map. Then $\Phi$ is strong 2-skew commutativity preserving if and only if $\Phi$ is the identity map.

Before the statement of our next corollary, let us recall some more notions. Denote by ${{\mathbb {F}}}$ the real field ${{\mathbb {R}}}$ or complex field ${{\mathbb {C}}}$. An indefinite inner product space means a linear space H over ${{\mathbb {F}}}$ equipped with a non-degenerate sesqui-linear Hermite functional $\langle \cdot , \cdot \rangle$. Let $(H, \langle \cdot , \cdot \rangle )$ be an indefinite inner product space. If there exist a positive subspace $H_+$ and a negative subspace $H_-$ such that $H = H_+ \oplus H_-$ and $(H_+, \langle \cdot , \cdot \rangle )$ is a Hilbert space when $\langle \cdot , \cdot \rangle$ is restricted to $H_+$, while $(H_-, -\langle \cdot , \cdot \rangle )$ is a Hilbert space when $-\langle \cdot , \cdot \rangle$ is restricted to $H_-$, we call H a complete indefinite inner product space or Krein space (Ref. [20]). ($H, \langle \cdot ,\cdot \rangle$) is a Krein space if and only if $(H, \langle \cdot ,\cdot \rangle _0)$ is a Hilbert space, where the inner product $\langle \cdot ,\cdot \rangle _0$ is defined by

$$\begin{aligned} \langle x,y\rangle _0=\left\{ \begin{array}{ll} \langle x,y\rangle &{}\quad \mathrm{if}\ x,y\in H_+; \\ -\langle x,y\rangle &{}\quad \mathrm{if}\ x,y\in H_-; \\ 0 &{}\quad \mathrm{if}\ x\in H_{\pm }, y\in H_{\mp }. \end{array}\right. \end{aligned}$$

The set of all bounded linear operators on H with respect to the indefinite inner product $\langle \cdot ,\cdot \rangle$ is the same as ${{\mathcal {B}}}(H, \langle \cdot ,\cdot \rangle _0)$, which is still denoted by ${{\mathcal {B}}}(H)$. For $A\in {{\mathcal {B}}}(H)$, denote by $A^\dagger$ the indefinite adjoint of A, which is determined by $\langle x, A^\dagger y\rangle =\langle Ax,y\rangle$ for all $x,y\in H$. Clearly, when $\langle \cdot ,\cdot \rangle$ is an inner product itself, $A^\dag =A^*$, the usual adjoint operator of A. Then $A\mapsto A^\dag$ is clearly an involution on ${\mathcal B}(H)$. An indefinite self-adjoint standard operator algebra ${\mathcal {A}}$ on a real or complex Krein space H is a subalgebra of ${{\mathcal {B}}}(H)$ which contains the identity operator I and all finite rank operators, and satisfies $A\in {{\mathcal {A}}}\Rightarrow A^\dag \in {{\mathcal {A}}}$. Obviously, every indefinite self-adjoint standard operator algebra ${\mathcal {A}}$ is a unital involution prime algebra with involution $\dag$ and containing many nontrivial self-adjoint idempotents, of which ${{\mathcal {Z}}}_S({{\mathcal {A}}})={{\mathbb {R}}}I$. In addition, by [21], for standard operator algebras, Lemma 4.1 is valid if we replace “linearly independent over ${{\mathcal {C}}}$” and “linear combination over ${\mathcal {C}}$” simply by “linearly independent” and “linear combination.” Then, by a check the proof of Theorem 3.1 in Sect. 4, the following corollary is true.

Corollary 3.4

Let H be a real or complex Krein space with $\dim H\ge 2$ and ${\mathcal A}\subseteq {{\mathcal {B}}}(H)$ be an indefinite self-adjoint standard operator algebra. Assume that $\Phi :{{\mathcal {A}}} \rightarrow {{\mathcal {A}}}$ is a surjective map. Then $\Phi$ is strong 2-skew commutativity preserving, that is, $\Phi$ satisfies

$$\begin{aligned} \Phi (A)^2\Phi (B)-2\Phi (A)\Phi (B)\Phi (A)^\dag +\Phi (B)(\Phi (A)^\dag )^2=A^2B-2ABA^\dag +B(A^\dag )^2 \end{aligned}$$

for any $A,B\in {\mathcal {A}}$, if and only if $\Phi$ is the identity map.

Let H be a real or complex Hilbert space with inner product $\langle \cdot ,\cdot \rangle$. For any invertible self-adjoint operator $J\in {{\mathcal {B}}}(H), \langle \cdot ,\cdot \rangle _J$ defined by $\langle x,y\rangle _J=\langle Jx,y\rangle$ is an indefinite inner product and $(H,\langle \cdot ,\cdot \rangle _J)$ is a Krein space. For $A\in {{\mathcal {B}}}(H), A^\dag =J^{-1}A^*J$. If $A^\dag =A$, we say that A is J-self-adjoint. Furthermore, a standard operator algebra ${{\mathcal {A}}}\subseteq {{\mathcal {B}}}(H)$ is called a J-self-adjoint standard operator algebra if $A\in {{\mathcal {A}}}$ implies $A^\dag =J^{-1}A^*J\in {{\mathcal {A}}}$.

Corollary 3.4′ 1

Let H be a Hilbert space and $J\in {{\mathcal {B}}}(H)$ be an invertible self-adjoint operator. Let ${\mathcal {A}}$ be a J-self-adjoint standard operator algebra and $\Phi :{{\mathcal {A}}}\rightarrow {{\mathcal {A}}}$ a surjective map. Then, $\Phi$ satisfies

$$\begin{aligned}&\Phi (A)^2\Phi (B)-2\Phi (A)\Phi (B)J^{-1}\Phi (A)^*J+\Phi (B)J^{-1}(\Phi (A)^*)^2J\\&\quad =A^2B-2ABJ^{-1}A^*J+BJ^{-1}(A^*)^2J \end{aligned}$$

for any $A,B\in {\mathcal {A}}$, if and only if $\Phi$ is the identity map.

Particularly, we have

Corollary 3.5

Let ${{\mathcal {M}}}_n({\mathbb {F}})$ be the algebra of $n\times n$ matrices over the real or complex field ${\mathbb {F}}$ and $J\in {{\mathcal {M}}}_n({\mathbb {F}})$ an invertible self-adjoint matrix. Let $\Phi : {{\mathcal {M}}}_n({\mathbb {F}})\rightarrow {\mathcal M}_n({\mathbb {F}})$ be a surjective map. Then $\Phi$ satisfies

$$\begin{aligned}&\Phi (A)^2\Phi (B)-2\Phi (A)\Phi (B)J^{-1}\Phi (A)^*J+\Phi (B)J^{-1}(\Phi (A)^*)^2J\\&\quad =A^2B-2ABJ^{-1}A^*J+BJ^{-1}(A^*)^2J \end{aligned}$$

for any $A,B\in {\mathcal {A}}$, if and only if $\Phi$ is the identity map.

Given an orthonormal basis of a complex Hilbert space H, the transpose map $A\mapsto A^t$ with respect to the basis is an involution operation on ${{\mathcal {B}}}(H)$. A standard operator algebra ${{\mathcal {A}}}\subseteq {{\mathcal {B}}}(H)$ is said to be symmetric (with respect to this basis) if $A\in {\mathcal A}\Rightarrow A^t\in {{\mathcal {A}}}$. Thus, every symmetric standard operator algebra is a unital involution prime algebra with the involution being taken the transpose and, in this situation, ${{\mathcal {Z}}}_S({{\mathcal {A}}})={{\mathcal {Z}}}({\mathcal {A}})={\mathbb {C}}I$. By Theorem 3.1 and the reason mentioned before Corollary 3.4, we have

Corollary 3.6

Let H be a complex Hilbert space with $\dim H\ge 2$ and ${\mathcal A}\subseteq {{\mathcal {B}}}(H)$ be a symmetric standard operator algebra with respect to some given orthonormal basis of H. Assume that $\Phi :{{\mathcal {A}}} \rightarrow {{\mathcal {A}}}$ is a surjective map. Then $\Phi$ satisfies

$$\begin{aligned} \Phi (A)^2\Phi (B)-2\Phi (A)\Phi (B)\Phi (A)^t+\Phi (B)(\Phi (A)^t)^2=A^2B-2ABA^t+B(A^t)^2 \end{aligned}$$

for any $A,B\in {\mathcal {A}}$ if and only if there exists $\lambda \in \{1, e^{i\frac{2\pi }{3}}, e^{i\frac{4\pi }{3}}\}$ such that $\Phi (A) = \lambda A$ for all $A \in {{\mathcal {A}}}$.

Corollary 3.7

Let ${{\mathcal {M}}}_n({\mathbb {C}})$ be the algebra of $n\times n$ complex matrices, and let $\Phi : {\mathcal M}_n({\mathbb {C}})\rightarrow {{\mathcal {M}}}_n({\mathbb {C}})$ be a surjective map. Then $\Phi$ satisfies

$$\begin{aligned} \Phi (A)^2\Phi (B)-2\Phi (A)\Phi (B)\Phi (A)^t+\Phi (B)(\Phi (A)^t)^2=A^2B-2ABA^t+B(A^t)^2 \end{aligned}$$

for any $A,B\in {{\mathcal {M}}}_n({\mathbb {C}})$, if and only if there exists $\lambda \in \{1, e^{i\frac{2\pi }{3}}, e^{i\frac{4\pi }{3}}\}$ such that $\Phi (A) = \lambda A$ for all $A \in {{\mathcal {M}}}_n(\mathbb C)$.

Corollaries 3.6 and 3.7 also reveal that, in general, for a unital prime $*$-ring ${\mathcal {R}}, \lambda \in {{\mathcal {C}}}_S$ and $\lambda ^3=1$ may not imply $\lambda =1$. Hence the maps in Theorem 3.1 may not be the identity map.

4 Proof of Theorem 3.1: Char$({{\mathcal {R}}})\not =2$

In this section, we prove Theorem 3.1 for the case when the characteristic of ${\mathcal {R}}$ is not 2. To do this, we need three lemmas.

The first lemma comes from ( [22, Theorem A.7]).

Lemma 4.1

Let ${{\mathcal {R}}}$ be a prime ring, and let $A_i, B_i, C_j, D_j\in {{\mathcal {R}}}, i=1,2,\ldots , n$ and $j=1,2,\ldots , m$, such that $\sum {{}_{i=1}^n} A_{i}T B_{i} = \sum {{}_{j=1}^m} C_{j}TD_{j}$ for all $T\in {{\mathcal {R}}}$. If $A_{1},\ldots ,A_{n}$ are linearly independent over ${\mathcal {C}}$ (the extended centroid of ${\mathcal {R}}$), then each $B_{i}$ is a linear combination of $D_{1},\ldots ,D_{m}$ over ${\mathcal {C}}$. Similarly, if $B_{1},\ldots ,B_{n}$ are linearly independent over ${\mathcal {C}}$, then each $A_{i}$ is a linear combination of $C_{1},\ldots ,C_{m}$ over ${\mathcal {C}}$. In particular, if $ATB = BTA$ for all $T\in {{\mathcal {A}}}$, then A and B are linearly dependent over ${\mathcal {C}}$.

The next two lemmas give some interesting properties possessed by 2-skew commutators.

Lemma 4.2

Let ${\mathcal {R}}$ be a unital prime $*$-ring with involution $*$ and containing a nontrivial symmetric idempotent. Then, for $S\in {{\mathcal {R}}}, {{}_*[X,S]_2}=0$ for all $X\in {\mathcal {R}}$ if and only if $S=0$.

Proof

The “if” part is obvious. Let us check the “only if ” part.

Let P be a nontrivial symmetric idempotent in ${\mathcal {R}}$. From ${}_*[PX,S]_2(I-P)=0$, we get $PXPXS(I-P)=0$ holds for all $X\in {{\mathcal {R}}}$. In particular, with $X=P$, we have $PS(I-P)=0$. Also, substituting $X+P$ we find

$$\begin{aligned} PXPS(I-P)+PXS(I-P)=PX(P+I)S(I-P)=0, \end{aligned}$$

so that $(P+I)S(I-P)=0$ since ${\mathcal {R}}$ is prime and $P\not =0$. Thus one gets $S(I-P)=(P+I)S(I-P)-PS(I-P)=0$. By an analogous process, starting from ${}_*[(I-P)X,S]_2P=0$, one finds $SP=0$. Hence $S=SP+S(I-P)=0$, as desired. $\square$

Lemma 4.3

Let ${\mathcal {R}}$ be a prime $*$-ring of characteristic not 2 and $S\in {{\mathcal {R}}}$. Then ${}_*[S,X]_2=0$ for any $X\in {\mathcal {R}}$ if and only if $S\in {\mathcal {Z}}_{S}({\mathcal {R}})$.

Proof

To check the “if” part, assume that $S\in \mathcal {Z}_{S}({\mathcal {R}})$. It is easily seen that

$$\begin{aligned} {}_*[S,X]_2 = S^{2}X-2SXS^{*}+X{S^{*}}^{2}=(S-S)^{2}X=0 \end{aligned}$$

holds for all $X\in {\mathcal {R}}.$

Next we check the “only if” part.

Assume that $S\not =0$ satisfies that

$$\begin{aligned} {}_*[S,X]_2 = S^{2}X-2SXS^{*}+X{S^{*}}^{2} = 0 \end{aligned}$$

for all $X \in {\mathcal {R}}$; that is,

$$\begin{aligned} S^{2}X+X{S^{*}}^{2}=2SXS^{*} \end{aligned}$$

(4.1)

for all $X \in {\mathcal {R}}$.

By Eq. (4.1) and Lemma 4.1, either $\{I,S^2\}$ is linearly dependent over ${\mathcal {C}}$ (i.e., $S^2\in {{\mathcal {Z}}}({{\mathcal {R}}})$) or in particular there exists $\alpha \in {{\mathcal {C}}}$ such that $I=\alpha S^*$ ($\alpha \not =0$), so that $S={\alpha ^{-1}}^* I\in {\mathcal Z}({{\mathcal {R}}})$. In any case, we have $S^2\in {\mathcal Z}({{\mathcal {R}}})$. Thus Eq. (4.1) can be rewritten as $X(S^2+{S^*}^2)=SX2S^*$. Again by Lemma 4.1, either $2S^*=0$ (and $S=0$ because char$({{\mathcal {R}}})\not =2$) or $\{2S^*\}$ is linearly independent over ${\mathcal {C}}$ and there exists $\beta \in {\mathcal {C}}$ such that $S=\beta I$. In any case, $S\in {{\mathcal {Z}}}({{\mathcal {R}}})$. Lastly, by Eq. (4.1) again, $(S^2-2SS^*+{S^*}^2)X=(S-S^*)^2X=(S-S^*)X(S-S^*)=0$ for every $X\in {\mathcal {R}}$, which implies, since ${\mathcal {R}}$ is prime, that $S-S^*=0$, i.e., $S\in {{\mathcal {Z}}}_S({\mathcal {R}})$. $\square$

Now we are at a position to give our proof of the main theorem for the case that the ring is 2-torsion-free.

Proof of Theorem 3.1:

Char$({\mathcal {R}})\not =2$. The “if” part of the theorem is obvious. In the sequel, we always assume that Char$({\mathcal {R}})\not =2$ and $\Phi :{\mathcal {R}} \rightarrow \mathcal {R}$ is a surjective strong 2-skew commutativity preserving map. We will check the “only if” part by several steps.

Step 1.$\Phi$ is additive.

Note that, for any $A,B,C\in {{\mathcal {R}}}$, we have ${}_*[C,A+B]_2={{}_*[C,A]_2}+{{}_*[C,B]_2}$. Then,

$$\begin{aligned}&{{}_*[\Phi (C), \Phi (A + B) - \Phi (A) - \Phi (B)]_2}\\&\quad = {{}_*[\Phi (C), \Phi (A + B)]_2} - {{}_*[\Phi (C), \Phi (A)]_2} -{{}_*[\Phi (C), \Phi (B)]_2}\\&\quad = {{}_*[C, A+B]_2 -{{}_*[C, A]_2} - {{}_*[C, B]_2}}\\&\quad = 0. \end{aligned}$$

By the surjectivity of $\Phi$ and Lemma 4.2, the above equation implies $\Phi (A+B)=\Phi (A)+\Phi (B)$. So $\Phi$ is additive.

Step 2. For any nontrivial symmetric idempotent P and any $X\in {\mathcal {R}}$, we have

$$\begin{aligned} {}_*[\Phi (P),X]_2\in P{\mathcal {R}}(I-P)+(I-P)\mathcal {R}P. \end{aligned}$$

(4.2)

Let $P\in {\mathcal {R}}$ be a nontrivial symmetric idempotent. By the surjectivity of $\Phi$, for every $X\in {\mathcal {R}}$ there exists $Y\in {\mathcal {R}}$ such that $X=\Phi (Y)$. Hence ${}_*[\Phi (P),X]_2={{}_*[\Phi (P),\Phi (Y)]_2}={{}_*[P,Y]_2}\in {{\mathcal {R}}}_{12}+{{\mathcal {R}}}_{21}$, as it is easily checked.

Step 3. For any $A\in {{\mathcal {R}}}$, we have $\Phi (A^*)=\Phi (A)^*$.

Note that, for any $A,B\in {{\mathcal {R}}}$, we have

$$\begin{aligned} {{}_*[B,A]_2}^*= {{}_*[B,A^*]_2}. \end{aligned}$$

Given $A\in {{\mathcal {R}}}$, it follows that

$$\begin{aligned} {}_*[\Phi (B),\Phi (A)^*]_2{=}{{}_*[\Phi (B),\Phi (A)]_2}^*={{}_*[B,A]_2}^* ={{}_*[B,A^*]_2}{=}{{}_*[\Phi (B),\Phi (A^*)]_2} \end{aligned}$$

holds for all $B\in {\mathcal {R}}$. Since $\Phi$ is surjective, we see from the above equation that ${}_*[X, \Phi (A^*)-\Phi (A)^*]_2=0$ for all $X\in {\mathcal {R}}$. Then, applying Lemma 4.2 ensures that $\Phi (A^*)=\Phi (A)^*$.

Step 4. For any nontrivial symmetric idempotent $P\in {\mathcal {R}}$, there exist $\lambda , \mu \in {\mathcal {C}}_S$ such that $\Phi (P) = \lambda P + \mu$ with $\lambda \not =0$.

Let $P\in {\mathcal {R}}$ be any nontrivial symmetric idempotent. Write $P_1=P, P_2=I-P$ and ${{\mathcal {R}}}_{ij}=P_i{{\mathcal {R}}}P_j$. Then, there are $S_{ij}\in {{\mathcal {R}}}_{ij}$ so that $\Phi (P_1)=S_{11}+S_{12}+S_{21}+S_{22}$.

Taking $X = X_{11}\in {\mathcal {R}}_{11}$ and applying Step 2, we have

$$\begin{aligned} 0=P_2{{}_*[\Phi (P_1),X_{11}]}_2P_2 =-2P_2\Phi (P_1)X_{11}{\Phi (P_1)^*}P_2 =-2S_{21}X_{11}S_{21}^*, \end{aligned}$$

which implies that $S_{21}=0$ because ${\mathcal {R}}$ is prime and char${\mathcal {R}}\ne 2$. Similarly, $S_{12}=0$.

Hence one has $\Phi (P) = S_{11} + S_{22}$. Let us determine $S_{11}$ and $S_{22}$ below.

Taking $X= X_{11}\in {{\mathcal {R}}}_{11}$ and applying Step 2, we have

$$\begin{aligned} {}_*[ \Phi (P_1),X_{11}]_2 = S_{11}^2X_{11}-2S_{11}X_{11}S_{11}^*+X_{11}{S_{11}^*}^2=0 \end{aligned}$$

holds for all $X_{11}\in {\mathcal {R}}_{11}$. That is ${}_*[S_{11},X_{11}]_2=0$ for all $X_{11}\in {\mathcal {R}}_{11}$. By Lemma 4.3, $S_{11}\in {\mathcal {Z}}_S({\mathcal {R}}_{11})$ since ${{\mathcal {R}}}_{11}$ is prime of characteristic not 2. It follows that $S_{11}XP_1=P_1XS_{11}$ for all $X\in {\mathcal {R}}$. Then, by Lemma 4.1, there exists some $\lambda _1\in {\mathcal {C}}$ such that $S_{11}=\lambda _1P_1$. Since $S_{11}=S_{11}^*, (\lambda _1-\lambda _1^{*})P_1=S_{11}-S_{11}^*=0$, which entails that $\lambda _1=\lambda _1^{*}$. So $\lambda _1\in {\mathcal {C}}_S$. Analogously, there exists some $\lambda _2\in {\mathcal {C}}_S$ such that $S_{22}=\lambda _2P_2$. It follows that $\Phi (P_1)=\lambda _1P_1+\lambda _2P_2=\lambda P + \mu I$, where $\lambda =\lambda _1-\lambda _2,\ \mu =\lambda _2$.

Finally, we still need to prove that $\lambda \ne 0$. On the contrary, we have $\Phi (P_1)= \mu I\in {\mathcal {Z}}_S$, which implies $X_{12}+X_{21}={{}_*[P_1, X]_2}={{}_*[\Phi (P_1),\Phi (X)]_2}=0$ for any $X=X_{11}+X_{12}+X_{21}+X_{22}\in {\mathcal {R}}$, contradicting the primeness of ${\mathcal {R}}$. The proof of the step is completed.

Note that ${\mathcal {C}}$ is a field since ${\mathcal {R}}$ is prime. So $\lambda \in {\mathcal {C}}$ is invertible. In the sequel, denote $\alpha =\lambda ^{-1}$ and fix a nontrivial symmetric idempotent $P_1$ and let $P_2=I-P_1$.

Step 5. For any $X_{ij}\in {\mathcal {R}}_{ij}$ with $1\le i\ne j\le 2$, we have $\Phi (X_{ij})=\alpha ^2 X_{ij}$.

For any $X_{12}\in {\mathcal {R}}_{12}$, write $\Phi (X_{12})= S_{11}+S_{12}+S_{21}+S_{22}$ with $S_{ij}\in {{\mathcal {R}}}_{ij}$. Then

$$\begin{aligned} X_{12}=\,&{{}_*[P_1,X_{12}]_2}\!=\!{{}_*[\Phi (P_1),\Phi (X_{12})]_2}\!=\! {{}_*[\lambda P_1\!+\!\mu , \Phi (X_{12})]_2}={{}_*[\lambda P_1,\Phi (X_{12})]_2}\\ =\,&\lambda ^2 P_1 \Phi (X_{12})-2\lambda P_1\Phi (X_{12})(\lambda P_1)+\Phi (X_{12})(\lambda ^2 P_1)\\=\,&\lambda ^2(S_{12}+S_{21})\in {\mathcal {R}}_{12}. \end{aligned}$$

Hence we get $\lambda ^2S_{12}=X_{12}$ and $\lambda ^2S_{21}=0$, which implies that $S_{21}=0$ and $S_{12}=(\lambda ^2)^{-1}X_{12}=\alpha ^2 X_{12}$. It follows that $\Phi (X_{12})=S_{11}+\alpha ^2 X_{12}+S_{22}$.

For any $Y=Y_{11}+Y_{21}+Y_{22}\in {\mathcal {R}}$, by the surjectivity of $\Phi$, there exists an element $T=T_{11}+T_{12}+T_{21}+T_{22}\in {\mathcal {R}}$ such that $\Phi (T)=Y$. So we have

$$\begin{aligned} {{}_*[\Phi (X_{12}),Y]_2}={{}_*[X_{12},T]_2}=-2X_{12}T_{22}X_{12}^*. \end{aligned}$$

(4.3)

Multiplying by $P_2$ from both sides of Eq. (4.3) gives

$$\begin{aligned} P_2{{}_*[\Phi (X_{12}),Y]_2}P_2=P_2{{}_*[X_{12},T]_2}P_2=0. \end{aligned}$$

Thus, a simple computation reveals

$$\begin{aligned} S_{22}^2Y_{22}-2S_{22}Y_{22}S_{22}^*+Y_{22}{S_{22}^*}^2=P_2{{}_*[\Phi (X_{12}),Y]_2}P_2=0, \end{aligned}$$

which implies that

$$\begin{aligned} _*[S_{22}, Y_{22}]_2=0 \end{aligned}$$

holds for any $Y_{22}\in {\mathcal {R}}_{22}$. So, by Lemma 4.3, $S_{22}\in \mathcal {Z}_S({\mathcal {R}}_{22})$ and hence, $S_{22}=\mu _2 P_2$ for some $\mu _2\in {\mathcal {C}}_S$.

Take $Y=Y_{21}$. Multiplying by $P_2$ from the left side and $P_1$ from the right of Eq. (4.3) gives

$$\begin{aligned} S_{22}^2Y_{21}+Y_{21}( S^*_{11})^2-2S_{22}Y_{21}S_{11}^*=0, \end{aligned}$$

which implies that $Y_{21}(S_{11}^*-\mu _2 P_1)^2=0$ holds for all $Y_{21}\in {\mathcal R}_{21}$. By the primeness of ${\mathcal {R}}$, we get $(S_{11}^*-\mu _2 P_1)^2=0$.

In summary, we have shown that, for any $X_{12}\in {\mathcal R}_{12}$, there exists $\mu (X_{12})\in {{\mathcal {C}}}_S$ such that

$$\begin{aligned} \Phi (X_{12})=\alpha ^2 X_{12}+S_{11}(X_{12})+\mu (X_{12})P_2, \end{aligned}$$

(4.4)

where $S_{11}(X_{12})\in {{\mathcal {R}}}_{11}$ satisfies that

$$\begin{aligned} (S_{11}(X_{12})-\mu (X_{12})P_1)^2=0 \end{aligned}$$

(4.5)

In a similar way, one can show that, for any $X_{21}\in {\mathcal R}_{21}$, there exists $\mu (X_{21})\in {{\mathcal {C}}}_S$ such that

$$\begin{aligned} \Phi (X_{21})=\alpha ^2 X_{21}+\mu (X_{21})P_1+S_{22}(X_{21}), \end{aligned}$$

(4.6)

where $S_{22}(X_{21})\in {{\mathcal {R}}}_{22}$ satisfies that

$$\begin{aligned} (S_{22}(X_{21})-\mu (X_{21})P_2)^2=0 \end{aligned}$$

(4.7)

Now, by Step 3, for any $X_{12}\in {{\mathcal {R}}}_{12}$, Eqs. (4.4)–(4.7) entail that

$$\begin{aligned}&\alpha ^2X_{12}^*+(S_{11}(X_{12}))^*+\mu (X_{12})P_2=\Phi (X_{12})^*\\&\quad =\Phi (X_{12}^*)= \alpha ^2 X_{12}^*+\mu (X_{12}^*)P_1+S_{22}(X_{12}^*). \end{aligned}$$

It follows that $(S_{11}(X_{12}))^*=\mu (X_{12}^*)P_1$ and consequently, by Eq. (4.5), $S_{11}(X_{12})=\mu (X_{12}^*)P_1=\mu (X_{12})P_1$. So

$$\begin{aligned} \Phi (X_{12})=\alpha ^2 X_{12} + \mu (X_{12}) P_1+\mu (X_{12}) P_2=\alpha ^2 X_{12} + \mu (X_{12}). \end{aligned}$$

By now we have shown that, for any $X_{12}\in {{\mathcal {R}}}_{12}$, there exists $\mu (X_{12})\in {{\mathcal {C}}}_S$ such that $\Phi (X_{12})=\alpha ^2 X_{12}+\mu (X_{12})$. Next we prove that $\mu (X_{12})=0$ holds for all $X_{12}$.

For any $Y_{12}\in {{\mathcal {R}}}_{12}$, we have

$$\begin{aligned} {}_*[P_1+Y_{12},X_{12}]_2=X_{12}-X_{12}Y_{12}^* \end{aligned}$$

and, by Step 1 and Step 4,

$$\begin{aligned}&X_{12}-X_{12}Y_{12}^* \\&\quad = {}_*[\Phi (P_1\!+\!Y_{12}),\Phi (X_{12})]_2={{}_*[\lambda P_1+\mu \!+\!\alpha ^2 Y_{12}+\mu (Y_{12}), \alpha ^2 X_{12}\!+\!\mu (X_{12})]_2}\\&\quad ={}_*[\lambda P_1+\alpha ^2 Y_{12}, \alpha ^2 X_{12}]_2+{{}_*[\lambda P_1+\alpha ^2 Y_{12}, \mu (X_{12})]_2}\\&\quad = \alpha ^2[(\lambda ^2P_1+\lambda \alpha ^2 Y_{12})X_{12}-2(\lambda P_1+\alpha ^2 Y_{12})X_{12}(\lambda P_1+\alpha ^2 Y_{12}^*)\\&\qquad +\,X_{12}(\lambda ^2P_1+\lambda \alpha ^2 Y_{12}^*)]\\&\qquad +\,\mu (X_{12})[(\lambda ^2P_1+\lambda \alpha ^2 Y_{12})-2(\lambda P_1+\alpha ^2 Y_{12})(\lambda P_1+\alpha ^2 Y_{12}^*)\\&\qquad +\,(\lambda ^2P_1+\lambda \alpha ^2 Y_{12}^*)]\\&\quad = \alpha ^2\lambda (\lambda X_{12}-\alpha ^2 X_{12}Y_{12}^*)+\mu (X_{12})(\alpha ^2\lambda Y_{12}+\alpha ^2\lambda Y_{12}^*-2\alpha ^4Y_{12}Y_{12}^*). \end{aligned}$$

This implies that $\mu (X_{12})\alpha ^2\lambda Y_{12}^*=0$ for any $Y_{12}\in {{\mathcal {R}}}_{12}$, which forces $\mu (X_{12})=0$. Therefore, $\Phi (X_{12})=\alpha ^2 X_{12}$ holds for each $X_{12}\in {{\mathcal {R}}}_{12}$.

Similarly, for any $X_{21}\in {{\mathcal {R}}}_{21}$ we have $\Phi (X_{21})=\alpha ^2 X_{21}$.

Step 6. For any $X_{ii}\in {\mathcal {R}}_{ii}, \Phi (X_{ii})=\lambda X_{ii}, i=1,2$.

Again, we only prove the assertion of Step 6 for the case $i=1$. The case $i=2$ is dealt with similarly.

For arbitrarily given $X_{11}\in {\mathcal {R}}$, write $\Phi (X_{11})=S_{11}+S_{12}+S_{21}+S_{22}$.

Since $0={{}_*[P_1, X_{11}]_2}={{}_*[\Phi (P_1), \Phi (X_{11})]_2}={{}_*[\lambda P_1+\mu , \Phi (X_{11})]_2}={{}_*[\lambda P_1, \Phi (X_{11})]_2}$, one can get $\lambda ^2(S_{12}+S_{21})=0$, which entails that $S_{12}=S_{21}=0$ and consequently, $\Phi (X_{11})=S_{11}+S_{22}$.

Then, for any $Y_{12}\in {\mathcal {R}}_{12}$, by Step 5, we have

$$\begin{aligned} \begin{array}{rl} 0=&{}{}_*{{[P_1+Y_{12}, X_{11}]}_2}={{}_*{[\Phi (P_1+Y_{12}), \Phi (X_{11})]}_2}\\ = &{} {{}_*[\lambda P_1+\mu +\alpha ^2Y_{12}, \Phi (X_{11})]_2}={{}_*[\lambda P_1+\alpha ^2 Y_{12}, S_{11}+S_{22}]_2}\\ =&{} (\lambda ^2P_1+\lambda \alpha ^2 Y_{12})(S_{11}+S_{22})-2(\lambda P_1+\alpha ^2 Y_{12})(S_{11}+S_{22})(\lambda P_1+\alpha ^2 Y_{12}^*)\\ &{} +\,(S_{11}+S_{22})(\lambda ^2P_1+\lambda \alpha ^2 Y_{12}^*)\\ =&{} \lambda \alpha ^2 Y_{12}S_{22}-2\alpha ^4Y_{12}S_{22}Y_{12}^*+\lambda \alpha ^2 S_{22}Y_{12}^*. \end{array} \end{aligned}$$

This implies that $\lambda \alpha ^2 Y_{12}S_{22}=0$ holds for any $Y_{12}\in {\mathcal {R}}_{12}$. By the primeness of ${\mathcal {R}}$, one sees that $S_{22}=0$.

Now, for any $Y_{21}\in {\mathcal {R}}_{21}$, we have

$$\begin{aligned}&Y_{21}X_{11}+X_{11}Y_{21}^*-2Y_{21}X_{11}Y_{21}^*={{}_*{{[P_1+Y_{21}, X_{11}]}_2}}\\&\quad ={{}_*{[\Phi (P_1+Y_{21}), \Phi (X_{11})]}_2}={{}_*[\lambda P_1+\alpha ^2Y_{21}, S_{11}]_2}\\&\quad =\lambda \alpha ^2 Y_{21}S_{11}+\lambda \alpha ^2S_{11}Y_{21}^*-2\alpha ^4Y_{21}S_{11}Y_{21}^*. \end{aligned}$$

This entails that $Y_{21}X_{11}=\lambda \alpha ^2 Y_{21}S_{11}$ holds for any $Y_{21}\in \mathcal {R}_{21}$. Recall that $\alpha =\lambda ^{-1}$. So one gets $P_2Y (S_{11}-\lambda X_{11})=0$ for all $Y\in {\mathcal {R}}$, and therefore, $S_{11}-\lambda X_{11}=0$ as ${\mathcal {R}}$ is prime. Consequently, $\Phi (X_{11})=\lambda X_{11}$ as desired.

Step 7.$\lambda ^3=I$ and $\Phi (A)= \lambda A$ for all $A\in {\mathcal {R}}$.

For any $X_{11},Y_{11}\in {{\mathcal {R}}}_{11}$, by Step 6 we have

$$\begin{aligned} {}_*[X_{11},Y_{11}]_2={{}_*[\Phi (X_{11}),\Phi (Y_{11})]_2}={{}_*[\lambda X_{11},\lambda Y_{11}]_2}=\lambda ^3{{}_*[X_{11},Y_{11}]_2} \end{aligned}$$

which ensures that $\lambda ^3=I$. Consequently, for any $X_{ij}\in {{\mathcal {R}}}_{ij}$ with $i\not =j$, by Step 5, $\Phi (X_{ij})=\alpha ^2 X_{ij}=\lambda X_{ij}$.

Then, for any $A\in {\mathcal {R}}$, writing $A=A_{11}+A_{12}+A_{21}+A_{22}$ with $A_{ij}\in {{\mathcal {R}}}_{ij}$, by Step 1, one gets

$$\begin{aligned} \begin{array}{rl} \Phi (A)=&{}\Phi (A_{11})+\Phi (A_{12})+\Phi (A_{21})+\Phi (A_{22})\\ = &{} \lambda (A_{11}+A_{12}+A_{21}+A_{22})=\lambda A. \end{array} \end{aligned}$$

The proof of Theorem 3.1 for the case char$({{\mathcal {R}}})\not =2$ is finished. $\square$

5 Proof of Theorem 3.1: Char$({{\mathcal {R}}})=2$

In this section, we always assume that the characteristic of the ring is 2. Note that Lemma 4.1 and Lemma 4.2 are still true for prime $*$-rings of characteristic 2, but Lemma 4.3 is not valid in this situation. What we can have is the following lemma.

Lemma 5.1

Let ${\mathcal {R}}$ be a prime $*$-ring of characteristic 2 and $S\in {{\mathcal {R}}}$. Then ${}_*[S,X]_2=0$ holds for any $X\in {\mathcal {R}}$ if and only if $S^2\in \mathcal {Z}_{S}({\mathcal {R}})$.

Proof

Since ${\mathcal {R}}$ has characteristic 2, ${}_*[ A,B]_2=A^2B+B(A^*)^2$. Thus if $S^2\in {\mathcal {Z}}_{S}(\mathcal {R})$, we always have

$$\begin{aligned} {}_*[S,X]_2=S^2X+X(S^*)^2=2XS^2=0. \end{aligned}$$

Conversely, assume that ${}_*[S,X]_2=S^2X+X(S^*)^2=0$ holds for all $X\in {\mathcal {R}}$. By Lemma 4.1, there exists $\alpha \in {\mathcal C}$, the extended centroid of ${\mathcal {R}}$, such that $S^2=\alpha I$. It follows that $X(\alpha I+(S^*)^2)=0$ for all X. By the primeness of ${\mathcal {R}}, (S^*)^2=-\alpha I=-S^2=S^2$. Thus we have $S^2\in {{\mathcal {Z}}}_S$. $\square$

Proof of Theorem 3.1:

Char$({{\mathcal {R}}})=2$. We only need to check the “only if” part.

Let P be a nontrivial symmetric idempotent in ${\mathcal {R}}$, and let $\Phi : {{\mathcal {R}}}\rightarrow {{\mathcal {R}}}$ be a map that is surjective and strong 2-skew commutativity preserving. It is easily checked that the assertions of Step 1–Step 3 in Sect. 4 are still valid for the case Char$({{\mathcal {R}}})=2$. So $\Phi$ is additive, $\Phi (A^*)=\Phi (A)^*$ and

$$\begin{aligned} {}_*[\Phi (P),X]_2\in P{\mathcal {R}}(I-P)+(I-P){\mathcal {R}}P \end{aligned}$$

holds for every $X\in {\mathcal {R}}$. We shall complete the remainder of the proof by several steps.

Step 1. There exist $\lambda , \mu \in {{\mathcal {C}}}_S$ such that $\Phi (P)^2=\lambda P+\mu$ with $\lambda \not =0$.

Again, write $P_1=P, P_2=I-P$ and ${{\mathcal {R}}}_{ij}=P_i{\mathcal R}P_j, i,j\in \{1,2\}$. Then $\Phi (P_1)^2=S_{11}+S_{12}+S_{21}+S_{22}$ for some $S_{ij}\in {{\mathcal {R}}}_{ij}$. Note that $\Phi (P_1)=\Phi (P_1)^*$ as $P_1^*=P_1$.

For any $X_{12}\in {{\mathcal {R}}}_{12}$, since

$$\begin{aligned} 0=P_2 {{}_*[\Phi (P_1), X_{12}]_2}P_2=P_2(\Phi (P_1)^2X_{12}+X_{12}(\Phi (P_1)^*)^2)P_2=P_2\Phi (P_1)^2X_{12}, \end{aligned}$$

we see that $S_{21} X_{12}=0$ holds for each $X_{12}$. Then it follows from the primeness of ${\mathcal {R}}$ we have $S_{21}=0$. Similarly, one can check that $S_{12}=0$. Hence $\Phi (P_1)^2=S_{11}+S_{22}$.

For any $X_{11}\in {{\mathcal {R}}}_{11}$, there exists $Y=Y_{11}+Y_{12}+Y_{21}+Y_{22}$ such that $\Phi (Y)=X_{11}$ as $\Phi$ is surjective. Then

$$\begin{aligned} Y_{12}+Y_{21}={{}_*[P_1,Y]_2}={{}_*[\Phi (P_1),X_{11}]_2}=S_{11}X_{11}+X_{11}S_{11}. \end{aligned}$$

This means that $S_{11}X_{11}+X_{11}S_{11}=0$. Therefore, for any $X_{11}$ we have

$$\begin{aligned} S_{11}=\lambda _1 P_1 \end{aligned}$$

for some $\lambda _1\in {{\mathcal {C}}}_S$ by Lemma 4.1. Similarly,

$$\begin{aligned} S_{22}=\mu P_2 \end{aligned}$$

for some $\mu \in {{\mathcal {C}}}_S$. Consequently,

$$\begin{aligned} \Phi (P_1)^2=\lambda P_1+\mu \end{aligned}$$

with $\lambda =\lambda _1-\mu \in {{\mathcal {C}}}_S$. Note that $\lambda \not =0$ because otherwise, by Lemma 5.1, $P_1X+XP_1={{}_*[P_1,X]_2}={{}_*[\Phi (P_1),\Phi (X)]_2}=0$ for every $X\in {{\mathcal {R}}}$, which is impossible by primeness.

Step 2.$\Phi ({{\mathcal {R}}}_{ii})\subset {\mathcal R}_{11}+{{\mathcal {R}}}_{22}$ and $\Phi ({{\mathcal {R}}}_{ij})\subset {{\mathcal {R}}}_{11}+{{\mathcal {R}}}_{ij}+{{\mathcal {R}}}_{22}$ for $i\not =j.$

For any $X\in {\mathcal {R}}$, since, by Step 1,

$$\begin{aligned} \begin{array}{rl} P_1X+XP_1=&{}{{}_*[P_1,X]_2}={{}_*[\Phi (P_1),\Phi (X)]_2}\\ =&{}(\lambda P_1+\mu )\Phi (X)+\Phi (X)(\lambda P_1+\mu )=\lambda (P_1\Phi (X)+\Phi (X)P_1),\end{array} \end{aligned}$$

we have

$$\begin{aligned} P_1XP_2=\lambda P_1\Phi (X)P_2\quad \mathrm{and}\quad P_2XP_1=\lambda P_2\Phi (X)P_1. \end{aligned}$$

(5.1)

Clearly, Eq. (5.1) implies that

$$\begin{aligned} \Phi ({{\mathcal {R}}}_{ii})\subset {{\mathcal {R}}}_{11}+{{\mathcal {R}}}_{22}\quad \mathrm{and}\quad \Phi ({{\mathcal {R}}}_{ij})\subset {{\mathcal {R}}}_{11}+{\mathcal R}_{ij}+{{\mathcal {R}}}_{22}\ \ \mathrm{for}\ i\not =j. \end{aligned}$$

Step 3.$\Phi (X_{ij})=\lambda ^{-1}X_{ij}$ for any $X_{ij}\in {{\mathcal {R}}}_{ij}$ with $i\not =j$.

If $X_{12}\in {{\mathcal {R}}}_{12}$, then, by Step 2, we have $\Phi (X_{12})=S_{11}+\lambda ^{-1}X_{12}+S_{22}$ for some $S_{ii}\in {{\mathcal {R}}}_{ii}, i=1,2$. We have to show that $S_{11}=S_{22}=0$. To do this, take any $W_{21}\in {{\mathcal {R}}}_{21}$ and take Y so that $\Phi (Y)=P_1+W_{21}$. By Step 2, it is easily checked that $Y\in {{\mathcal {R}}}_{11}+{{\mathcal {R}}}_{21}+{\mathcal R}_{22}$ and $P_2{{}_*[Y,X_{12}]_2}P_1=0$. Therefore, we have

$$\begin{aligned} \begin{array}{rl} 0=&{}P_2{{}_*[\Phi (Y),\Phi (X_{12})]_2}P_1=P_2{{}_*[P_1+W_{21},S_{11}+\lambda ^{-1}X_{12}+S_{22}]_2}P_1\\ =&{} P_2((P_1+W_{21})(S_{11}+\lambda ^{-1}X_{12}+S_{22})+(S_{11}+\lambda ^{-1}X_{12}+S_{22})(P_1+W_{21}^*))P_1\\ =&{}W_{21}S_{11}. \end{array} \end{aligned}$$

As $W_{21}$ is arbitrary, the primeness of ${\mathcal {R}}$ implies that $S_{11}=0$. Similarly, considering $P_1+W_{12}$ gives $S_{22}=0$. Hence $\Phi (X_{12})=\lambda ^{-1}X_{12}$ holds for every $X_{12}\in {{\mathcal {R}}}_{12}$.

The case for $X_{21}\in {{\mathcal {R}}}_{21}$ can be dealt with similarly.

Step 4. There exists $\delta \in {{\mathcal {C}}}_S$ with $\delta ^3=I$ such that $\Phi (X_{ij})=\delta X_{ij}$ for each $X_{ij}\in {{\mathcal {R}}}_{ij}, i,j\in \{1,2\}$.

Write $\Phi (P_1)=S_{11}+S_{22}$ and note that $\Phi (P_1)^*=\Phi (P_1)$. For any $X_{11}\in {{\mathcal {R}}}_{11}$ and $X_{12}\in {{\mathcal {R}}}_{12}$, it follows from Step 2 and Step 3 that $\Phi (X_{11})=A_{11}+A_{22}$ for some $A_{ii}\in {\mathcal R}_{ii}, i=1,2$ and

$$\begin{aligned}\begin{array}{rl} 0=&{}P_2{{}_*[P_1+X_{12},X_{11}]_2}P_1=P_2{{}_*[\Phi (P_1)+\Phi (X_{12}),\Phi (X_{11})]_2}P_1\\ =&{}P_2(\Phi (P_1) +\lambda ^{-1}X_{12})^2\Phi (X_{11})+\Phi (X_{11})(\Phi (P_1)+\lambda ^{-1}X_{12}^*)^2)P_1\\ =&{}P_2(\lambda P_1+\mu +\lambda ^{-1}(S_{11}+S_{22})X_{12}+\lambda ^{-1}X_{12}(S_{11}+S_{22}))(A_{11}+A_{22})P_1\\ &{} +\,P_2(A_{11}+A_{22})(\lambda P_1+\mu + \lambda ^{-1}X_{12}^*(S_{11}+S_{22})+\lambda ^{-1}(S_{11}+S_{22})X_{12}^*)P_1\\ =&{} \lambda ^{-1}(A_{22}X_{12}^*S_{11}+A_{22}S_{22}X_{12}^*). \end{array} \end{aligned}$$

Consequently,

$$\begin{aligned} S_{11}X_{12}A_{22}^*=X_{12}S_{22}A_{22}^* \end{aligned}$$

(5.2)

holds for all $X_{12}\in {{\mathcal {R}}}_{12}$. Taking $X_{11}=P_1$ in Eq. (5.2) and applying Lemma 4.1 again, since $P_1\not =0$, one gets

$$\begin{aligned} S_{22}^2=\alpha S_{22} \end{aligned}$$

for some $\alpha \in {{\mathcal {C}}}_S$. Suppose $S_{22}\not =0$ to reach a contradiction; then, we get $S_{11}=\alpha P_1$ and $\alpha A_{22}^*=S_{22}A_{22}^*$ from Eq. (5.2) by primeness. Clearly, $\alpha \not =0$ as $\alpha ^2 P_1=S_{11}^2=\lambda P_1\not =0$. Since $S_{22}\not =0, \mu \not =0$. Then it follows from the above equations that $S_{22}=\alpha ^{-1}\mu P_2=\alpha P_2$. Thus we have $\Phi (P_1)=S_{11}+S_{22}=\alpha$ which implies that $\mu =\Phi (P_1)^2=\lambda P_1+\mu$, a contradiction since $\lambda \not =0$. So $S_{22}=0$. Due to $S_{11}^2+S_{22}^2=\Phi (P_1)^2=(\lambda +\mu ) P_1+\mu P_2$, we get $\mu =0$ and $S_{11}\not =0$. Then, it follows from Eq. (5.2) we get $S_{11}X_{12}A_{22}^*=0$ for every $X_{12}$, which forces $A_{22}=0$. Therefore, $\Phi (X_{11})=A_{11}$ for every $X_{11}\in {{\mathcal {R}}}_{11}$.

Similarly, for every $X_{22}\in {{\mathcal {R}}}_{22}$ we can show that $\Phi (X_{22})=B_{22}\in {{\mathcal {R}}}_{22}$.

Now, for any $X_{12}\in {{\mathcal {R}}}_{12}$ and $X_{22}\in {\mathcal R}_{22}$ we have

$$\begin{aligned}&X_{12}X_{22}+X_{22}X_{12}^*={{}_*[P_1+X_{12}, X_{22}]_2}\\&\quad ={}_*[\Phi (P_1)+\Phi (X_{12}), \Phi (X_{22})]_2={{}_*[S_{11}+\lambda ^{-1}X_{12}, B_{22}]_2}\\&\quad = (S_{11}+\lambda ^{-1}X_{12})^2B_{22}+B_{22}(S_{11}+\lambda ^{-1}X_{12}^*)^2\\&\quad = \lambda ^{-1}(S_{11}X_{12}B_{22}+B_{22}X_{12}^*S_{11}), \end{aligned}$$

which implies that

$$\begin{aligned} \lambda ^{-1}S_{11}X_{12}B_{22}=X_{12}X_{22} \end{aligned}$$

(5.3)

holds for all $X_{12}\in {{\mathcal {R}}}_{12}$. Put $\Phi (P_2)=T_{22}$. Since $\lambda ^{-1}T_{22}\not =0$, taking $X_{22}=P_2$ in Eq. (5.3) we find by Lemma 4.1 that

$$\begin{aligned} S_{11}=\delta P_1 \end{aligned}$$

for some $\delta \in {{\mathcal {C}}}_S$. In addition, by primeness we get $\delta ^2=\lambda$ from $\delta ^2P_1=S_{11}^2=\lambda P_1$.

Moreover, for any $X_{21}\in {{\mathcal {R}}}_{21}$,

$$\begin{aligned} X_{21}X_{11}+X_{11}X_{21}^*= & {} {{}_*[P_1+X_{21},X_{11}]_2}\\= & {} {{}_*[\Phi (P_1+X_{21}),\Phi (X_{11})]_2} = {{}_*[\delta P_1+\lambda ^{-1}X_{21}, A_{11}]_2}\\= & {} \delta \lambda ^{-1}(X_{21}A_{11}+A_{11}X_{21}^*)=\delta ^{-1}(X_{21}A_{11}+A_{11}X_{21}^*). \end{aligned}$$

It follows that $X_{21}X_{11}=\delta ^{-1}X_{21}A_{11}$ for all $X_{21}$ and consequently, $A_{11}=\delta X_{11}$.

In addition, because

$$\begin{aligned} {}_*[X_{11},Y_{11}]_2={{}_*[\Phi (X_{11}),\Phi (Y_{11})]_2}={{}_*[\delta X_{11},\delta Y_{11}]_2}=\delta ^3{{}_*[X_{11},Y_{11}]_2} \end{aligned}$$

holds for any $X_{11},Y_{11}\in {{\mathcal {R}}}_{11}$, we see that $\delta ^3=I$. Thus we have $\lambda ^{-1}=\delta ^{-2}=\delta$ and $\Phi (X_{ij})=\lambda ^{-1}X_{ij}=\delta X_{ij}$ holds for any $X_{ij}$ with $i\not =j$.

Analogously, the consideration of ${}_*[P_1+X_{12},X_{22}]_2$ gives $\Phi (X_{22})=\delta X_{22}$ for any $X_{22}\in {{\mathcal {R}}}_{22}$.

Step 5.$\Phi (A)=\delta A$ for any $A\in {{\mathcal {R}}}$.

This follows directly from Step 4 by additivity.

This completes the proof of Theorem 3.1 for the case char$({{\mathcal {R}}})=2$. $\square$