Abstract
Let \(\mathcal{A}\) be a prime \(\ast\)-algebra. In this paper, assuming that \(\Phi:\mathcal{A}\to\mathcal{A}\) satisfies \(\Phi(A\diamond B \diamond C)=\Phi(A)\diamond B \diamond C+A\diamond\Phi(B) \diamond C+A \diamond B \diamond \Phi(C)\) where \(A\diamond B = A^{*}B + B^{*}A\) for all \(A,B\in\mathcal{A}\), we prove that \(\Phi\) is additive an \(\ast\)-derivation.
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1. INTRODUCTION
Let \(\mathcal{R}\) be a \(*\)-algebra. For \(A,B\in\mathcal{R}\), we write \(A\bullet B=AB+BA^{*}\) and \([A,B]_{*}=AB-BA^{*}\) for the \(\ast\)-Jordan product and \(\ast\)-Lie product, respectively. These products play an important role in some research topics, and their study has recently attracted the attention of many authors (for example, see [1]–[5]).
Recall that a map \(\Phi:\mathcal{R}\to\mathcal{R}\) is said to be an additive derivation if
for all \(A,B\in\mathcal{R}\). A map \(\Phi\) is an additive \(\ast\)-derivation if it is an additive derivation and \(\Phi(A^{*})=\Phi(A)^{*}\). Derivations are very important maps both in theory and applications and have been studied intensively ([6]–[11]).
A von Neumann algebra\(\mathcal{A}\) is a self-adjoint subalgebra of \(B(H)\), the algebra of all bounded linear operators acting on a complex Hilbert space, which satisfies the double commutant property: \(\mathcal{A}^{''}=\mathcal{A}\) where \(\mathcal{A}^{'}=\{T\in B(H), TA=AT\}\) for all \(A\in\mathcal{A}\), and \(\mathcal{A}^{''}=\{\mathcal{A}^{'}\}^{'}\). We denote by \(\mathcal{Z}(\mathcal{A})=\mathcal{A}^{'}\cap \mathcal{A}\) the center of \(\mathcal{A}\). A von Neumann algebra \(\mathcal{A}\) is called a factor if its center is trivial, i.e., \(\mathcal{Z}(\mathcal{A})=\mathbb{C}I\). For \(A\in\mathcal{A}\), recall that the central carrier of \(A\), denoted by \(\overline{A}\), is the smallest central projection \(P\) such that \(PA=A\). It is not difficult to see that \(\overline{A}\) is the projection onto the closed subspace spanned by \(\{BAx : B\in \mathcal{A}, x\in H\}\). If \(A\) is self-adjoint, then the core of \(A\), denoted by \(\underline{A}\), is \(\sup\{S\in\mathcal{Z}(\mathcal{A}): S=S^{*}, S\leq A\}\). If \(A=P\) is a projection, it is clear that \(\underline{P}\) is the largest central projection \(Q\) satisfying \(Q\leq P\). A projection \(P\) is said to be core-free if \(\underline{P}=0\) (see [12]). It is easy to see that \(\underline{P}=0\) if and only if \(\overline{I-P}=I\), [13]-[14].
Recently, Yu and Zhang in [15] proved that every nonlinear \(\ast\)-Lie derivation from a factor von Neumann algebra into itself is an additive \(\ast\)-derivation. Also, in [16], Li, Lu, and Fang investigated nonlinear \(\lambda\)-Jordan \(\ast\)-derivations. They showed that if \(\mathcal{A}\subseteq\mathcal{B(H)}\) is a von Neumann algebra without central Abelian projections and \(\lambda\) is a nonzero scalar, then \(\Phi:\mathcal{A} \longrightarrow \mathcal{B(H)}\) is a nonlinear \(\lambda\)-Jordan \(\ast\)-derivation if and only if \(\Phi\) is an additive \(\ast\)-derivation.
On the other hand, many mathematicians have studied the \(\ast\)-Jordan product \(A\bullet B=AB+BA^{*}\). In [17], F. Zhang proved that every nonlinear \(\ast\)-Jordan derivation map \(\Phi:\mathcal{A}\to\mathcal{A}\) on a factor von Neumann algebra is an additive \(\ast\)-derivation.
In [18], we showed that \(\ast\)-Jordan derivation map on every factor von Neumann algebra \(\mathcal{A} \subseteq \mathcal{B(H)}\) is an additive \(\ast\)-derivation.
Quite recently, the authors of [19] discussed some bijective maps preserving the new product \(A^{*}B+B^{*}A\) between von Neumann algebras with no central Abelian projections. In other words, they considered the map \(\Phi\) that satisfies the following assumption:
They showed that such a map is the sum of a linear \(\ast\)-isomorphism and a conjugate linear \(\ast\)-isomorphism.
We say that \(\mathcal{A}\) is prime, i.e., if \(A\mathcal{A}B=\lbrace0\rbrace\) for \(A,B \in \mathcal{A}\), then \(A = 0\) or \(B = 0\).
In [20], we assumed that \(\mathcal{A}\) is a prime \(\ast\)-algebra and the map \(\Phi:\mathcal{A}\to\mathcal{A}\) satisfies the following condition:
where \(A\diamond B = A^{*}B + B^{*}A\) for all \(A,B\in\mathcal{A}\). We proved that, in this case, \(\Phi\) is an additive \(\ast\)-derivation.
The authors of [21] introduced the concept of \(\ast\)-Lie triple derivations. A map \(\Phi:\mathcal{A}\to\mathcal{A}\) is a nonlinear\(\ast\)-Lie triple derivation if
for all \(A, B, C\in\mathcal{A}\), where \([A,B]_{\ast}=AB-BA^{*}\). They showed that if \(\Phi\) preserves the above characterization of factor von Neumann algebras, then \(\Phi\) is an additive \(\ast\)-derivation.
Motivated by the above results, we introduce the triple product \(A\diamond B\diamond C:=(A\diamond B) \diamond C\), where \(A\diamond B = A^{*}B + B^{*}A\). In this paper, let \(\mathcal{A}\) be a prime \(\ast\)-algebra, and let \(\Phi:\mathcal{A}\to\mathcal{A}\) satisfy the following equality:
for all \(A,B,C\in\mathcal{A}\). We prove that \(\Phi\) is an additive \(\ast\)-derivation.
2. MAIN RESULTS
Our main theorem is as follows.
Theorem 1.
Let \(\mathcal{A}\) be a prime \(\ast\) -algebra, and let \(\Phi:\mathcal{A}\to \mathcal{A}\) satisfy the condition
for all \(A,B,C\in\mathcal{A}\) , then \(\Phi\) is an additive \(\ast\) -derivation.
Proof. Let \(P_{1}\) be a nontrivial projection in \(\mathcal{A}\), and let \(P_{2}=I_{\mathcal{A}}-P_{1}\). Denote \(\mathcal{A}_{ij}=P_{i}\mathcal{A}P_{j}\) for \(i,j=1,2\); then \(\mathcal{A}=\sum_{i,j=1}^{2}\mathcal{A}_{ij}\). For every \(A\in\mathcal{A}\), we can write \(A=A_{11}+A_{12}+A_{21}+A_{22}\). In what follows, when we write \(A_{ij}\), this will indicate that \(A_{ij}\in\mathcal{A}_{ij}\). In order to show additivity of \(\Phi\) on \(\mathcal{A}\), we apply the above partitions of \(\mathcal{A}\) and establish some claims that imply that \(\Phi\) is additive on each \(\mathcal{A}_{ij}\) for \(i,j=1,2\).
Thus, the above theorem is a consequence of the following claims.
Claim 1.
\(\Phi(0)=0\) .
This claim is easy to prove.
Claim 2.
\(\Phi({I}/{2})=0\) , \(\Phi(-{I}/{2})=0\) , and \(\Phi(i{I}/{2})=0\) .
To show that \(\Phi({I}/{2})=0\), we write
Thus,
From (2.2), we deduce that \(\Phi({I}/{2})\) is self-adjoint. Therefore, we have the desired result.
To prove that \(\Phi({I}/{2})=0\), we write
It follows that
Then
On the other hand, we have
It follows that
Then, from (2.4) and (2.5), we obtain \(\Phi(-\frac{I}{2})=0\). To show that \(\Phi(i\frac{I}{2})=0\), we write
Thus,
Also, we have
Thus,
From (2.6) and (2.7), we obtain \(\Phi(i\frac{I}{2})=0\).
Claim 3.
Suppose that, for each \(A \in \mathcal{A}\) ,
-
1.
\(\Phi(-iA)=-i\Phi(A).\)
-
2.
\(\Phi(iA)=i\Phi(A).\)
It is easy to see that
Thus,
It follows that
On the other hand, one can check that
Thus,
It follows that
Equivalently, we have
By adding equeations (2.8) and (2.10), we obtain
Similarly, we can show that \(\Phi(iA)=i\Phi(A)\).
Claim 4.
For each \(A_{11} \in \mathcal{A}_{11}\) , \(A_{12} \in \mathcal{A}_{12}\) , the following equality holds:
Setting \(T=\Phi(A_{11}+A_{12})-\Phi(A_{11})-\Phi(A_{12})\) let us prove that \(T=0\). We have
Since \(T_{11}+T_{12}+T_{21}+T_{22}\), it follows that
So \(T_{22}=T_{21}=0.\) Similarly, we have
Therefore, \(T\diamond C_{12} \diamond P_{1}=0\). So \(T_{11}^{*}C_{12}+C_{12}^{*}T_{11}=0\). It follows that \(T_{11}^{*}C_{12}=0\). Hence, for all \(C \in \mathcal{A}\), we have \(T_{11}^{*}C P_{2}=0\). Since \(\mathcal{A}\) is prime, it follows that \(T_{11}=0\). Similarly, we can show that \(T_{12}=0\) by applying \(P_{2}\) instead of \(P_{1}\) in the above.
Claim 5.
For each \(A_{11}\in\mathcal{A}_{11}, A_{12}\in\mathcal{A}_{12}, A_{21}\in\mathcal{A}_{21}\) , and \(A_{22}\in\mathcal{A}_{22}\) ,
-
1.
\(\Phi(A_{11}+A_{12}+A_{21})=\Phi(A_{11})+\Phi(A_{12})+\Phi(A_{21}).\)
-
2.
\(\Phi(A_{12}+A_{21}+A_{22})=\Phi(A_{12})+\Phi(A_{21})+\Phi(A_{22}).\)
Then
From Claim 4, we obtain
It follows that \(T\diamond C_{21} \diamond I=0\). Since \(T=T_{11}+T_{12}+T_{21}+T_{22}\), we have
Therefore, \(T_{22}=T_{21}=0\). From Claim 4, we obtain
So \(T \diamond P_{1} \diamond P_{1}=0\) Then \(2T_{11}+2T_{11}^{*}+T_{12}+T_{12}^{*}=0\). Therefore,
Using Claim 3 and Claim 4, we obtain
Thus, \(T \diamond iP_{1} \diamond I =0\). We obtain
Relations (2.11) and (2.12) imply \(T_{11}=0\). Similarly, we can show that
Claim 6.
For each \(A_{11}\in\mathcal{A}_{11}, A_{12}\in\mathcal{A}_{12}, A_{21}\in\mathcal{A}_{21}\) , and \(A_{22}\in\mathcal{A}_{22}\) ,
Then
From Claim 5, we obtain
Thus, \(T\diamond C_{12} \diamond I=0\). It follows that
Therefore, \(T_{11}=T_{12}=0\). Similarly, by applying \(C_{21}\) instead of \(C_{12}\) in the above, we obtain \(T_{21}=T_{22}=0\).
Claim 7.
For each \(A_{ij},B_{ij} \in \mathcal{A}_{i}\) such that \(i\neq j\) , the following equality holds:
It is easy to show that
Thus, we can write
Thus, we have shown that
By an easy computation, we obtain
Then, we have
We have shown that
From Claim 3 and the above equation, we have
By adding equations (2.13) and (2.14), we obtain
Claim 8.
For each \(A_{ii},B_{ii} \in \mathcal{A}_{ii}\) such that \(1\leq i \leq 2\) , the following equality holds:
Let us show that
We have
Therefore,
Thus, \(T_{ij}=T_{ji}=T_{jj}=0\).
On the other hand, for every \(C_{ij}\in\mathcal{A}_{ij}\), we have
Thus, \(T\diamond C_{ij} \diamond I=0\); then \(T_{ii} \diamond C_{ij} \diamond I=0\). We have \(T_{ii}^{*}C_{ij}+C_{ij}^{*}T_{ii}=0\). We know that if \(\mathcal{A}\) is prime, then \(T_{ii}=0\). Hence the additivity of \(\Phi\) follows from the above claims. □
In the rest of this paper, we show that \(\Phi\) is a \(\ast\)-derivation.
Claim 9.
\(\Phi\) preserves stars.
Since \(\Phi({I}/{2})=0\), we have
Therefore,
Thus, we have shown that \(\Phi\) preserves stars.
Claim 10.
\(\Phi\) is a derivation.
For every \(A,B\in\mathcal{A}\), we have
Therefore,
Also
So we have
By adding equations (2.15) and (2.16), we obtain
From (2.17), Claims 3 and 9, it follows that
Therefore,
From (2.17) and (2.18), we obtain
This completes the proof. □
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Funding
The research of the first author was supported by the Talented Young Scientist Program of the Ministry of Science and Technology of China (Iran-19-001).
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Darvish, V., Nouri, M. & Razeghi, M. Nonlinear Triple Product \(A^{*}B + B^{*}A\) for Derivations on \(\ast\)-Algebras. Math Notes 108, 179–187 (2020). https://doi.org/10.1134/S0001434620070196
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DOI: https://doi.org/10.1134/S0001434620070196