Abstract
Let \( {\mathcal{A}} \) be a unital \( \ast \)-algebra containing a nontrivial projection. Under some mild conditions on \( {\mathcal{A}} \), it is shown that a map \( \Phi:{\mathcal{A}}\rightarrow{\mathcal{A}} \) is a nonlinear mixed Jordan triple \( * \)-derivation if and only if \( \Phi \) is an additive \( * \)-derivation. In particular, we apply the above result to prime \( \ast \)-algebras, von Neumann algebras with no central summands of type \( I_{1} \), factor von Neumann algebras, and standard operator algebras.
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1. Introduction
Let \( {\mathcal{A}} \) be a \( * \)-algebra over the complex field \( {} \). Given \( A,B\in\mathcal{A} \), we call the product \( [A,B]_{\ast}=AB-BA^{\ast} \) the skew Lie product and \( A\bullet B=AB+BA^{\ast} \) the Jordan \( * \)-product. The two new products are fairly meaningful and important and have been studied by many authors (see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]). Recall that an additive map \( \Phi:\mathcal{A}\rightarrow{\mathcal{A}} \) is said to be an additive derivation if
for all \( A,B\in{\mathcal{A}} \). Furthermore, \( \Phi \) is said to be an additive \( * \)-derivation if \( \Phi \) is an additive derivation and satisfies \( \Phi(A^{*})=\Phi(A)^{*} \) for all \( A\in{\mathcal{A}} \). A map (without the additivity assumption) \( \Phi:{\mathcal{A}}\rightarrow{\mathcal{A}} \) is said to be a nonlinear Jordan \( * \)-derivation or a nonlinear skew Lie derivation if \( \Phi(A\bullet B)=\Phi(A)\bullet B+A\bullet\Phi(B) \) or \( \Phi([A,B]_{\ast})=[\Phi(A),B]_{\ast}+[A,\Phi(B)]_{\ast} \) for all \( A,B\in{\mathcal{A}} \). Many authors have paid more attentions on the problem about Jordan \( * \)-derivations, skew Lie derivations and triple derivations, such as Jordan triple \( * \)-derivations and skew Lie triple derivations (see [15,16,17,18,19,20,21,22,23,24]). For example, Taghavi et al. [24] investigated a nonlinear \( \lambda \)-Jordan triple \( * \)-derivation on prime \( * \)-algebras; i.e., for all \( A,B,C\in{\mathcal{A}} \),
where \( A\diamondsuit_{\lambda}B=AB+\lambda BA^{*} \) such that a complex scalar \( |\lambda|\neq 0,1 \), and \( \Phi \) is additive. Moreover, if \( \Phi(I) \) is self-adjoint, then \( \Phi \) is a \( * \)-derivation.
Recently, many authors have studied the isomorphisms and derivations corresponding to the new products of the mixture of Lie product and skew Lie product. For example, Yang and Zhang [25, 26] studied the nonlinear maps that preserve the mixed skew Lie triple product \( [[A,B]_{\ast},C] \) and \( [[A,B],C]_{\ast} \) on factor von Neumann algebras. Zhou, Yang, and Zhang [27] studied the structure of the nonlinear mixed Lie triple derivations on prime \( * \)-algebras. In this paper, we consider the derivations corresponding to the new product of the mixture of the skew Lie product and the Jordan \( * \)-product. A map \( \Phi:{\mathcal{A}}\rightarrow{\mathcal{A}} \) is said to be a nonlinear mixed Jordan triple \( * \)-derivation if
for all \( A,B,C\in{\mathcal{A}} \). Under some mild conditions on a \( * \)-algebra \( \mathcal{A} \), we prove that a map \( \Phi:{\mathcal{A}}\rightarrow{\mathcal{A}} \) is a nonlinear mixed Jordan triple \( * \)-derivation if and only if \( \Phi \) is an additive \( * \)-derivation. In particular, we apply the above result to prime \( \ast \)-algebras, von Neumann algebras with no central summands of type \( I_{1} \), factor von Neumann algebras, and standard operator algebras.
2. The Main Result and Its Proof
Our main result in this paper reads as follows:
Theorem 2.1
Let \( {\mathcal{A}} \) be a unital \( \ast \)-algebra with the unit \( I \). Assume that \( {\mathcal{A}} \) contains a nontrivial projection \( P \) that satisfies
and
Then a map \( \Phi:{\mathcal{A}}\rightarrow{\mathcal{A}} \) satisfies
for all \( A,B,C\in{\mathcal{A}} \) if and only if \( \Phi \) is an additive \( * \)-derivation.
Proof
Let \( P_{1}=P \) and \( P_{2}=I-P \). Put \( {\mathcal{A}}_{jk}=P_{j}{\mathcal{A}}P_{k} \), \( j,k=1,2 \). Then
In the sequel \( A_{jk} \) indicates that \( A_{jk}\in{\mathcal{A}}_{jk} \). Clearly, we only need to prove the necessity. We will complete the proof by several claims:
Claim 1
\( \Phi(0)=0 \).
Indeed,
Claim 2
\( \Phi \) is additive.
We will complete the proof of Claim 2 in several steps.
Step 2.1. Given \( A_{12}\in{\mathcal{A}}_{12} \) and \( B_{21}\in{\mathcal{A}}_{21} \), we have \( \Phi(A_{12}+B_{21})=\Phi(A_{12})+\Phi(B_{21}) \).
We only need show that \( T=\Phi(A_{12}+B_{21})-\Phi(A_{12})-\Phi(B_{21})=0 \). Since
where \( i \) is the imaginary unit; it follows from Claim 1 that
From this we get \( [I\bullet(i(P_{2}-P_{1})),T]_{*}=0 \). So \( T_{11}=T_{22}=0 \).
Since \( [I\bullet A_{12},P_{1}]_{*}=0 \), it follows that
Hence \( [I\bullet T,P_{1}]_{*}=0 \), from which we get that \( T_{21}=0 \). Similarly, we can show that \( T_{12}=0 \), proving the step.
Step 2.2. For all \( A_{11}\in{\mathcal{A}}_{11},B_{12}\in{\mathcal{A}}_{12} \), \( C_{21}\in{\mathcal{A}}_{21} \), and \( D_{22}\in{\mathcal{A}}_{22} \) we have
and
Let \( T=\Phi(A_{11}+B_{12}+C_{21})-\Phi(A_{11})-\Phi(B_{12})-\Phi(C_{21}) \).
It follows from Step 2.1 that
From this we get \( [I\bullet(iP_{2}),T]_{*}=0 \). So \( T_{12}=T_{21}=T_{22}=0 \).
Since
it follows that
from which we get \( [I\bullet(i(P_{2}-P_{1})),T]_{*}=0 \). So \( T_{11}=0 \), and then \( T=0 \). Similarly, \( \Phi(B_{12}+C_{21}+D_{22})=\Phi(B_{12})+\Phi(C_{21})+\Phi(D_{22}) \).
Step 2.3. For all \( A_{11}\in{\mathcal{A}}_{11} \), \( B_{12}\in{\mathcal{A}}_{12} \), \( C_{21}\in{\mathcal{A}}_{21} \), and \( D_{22}\in{\mathcal{A}}_{22} \), we have
Let \( T=\Phi(A_{11}+B_{12}+C_{21}+D_{22})-\Phi(A_{11})-\Phi(B_{12})-\Phi(C_{21})-\Phi(D_{22}) \). It follows from Step 2.2 that
From this we get \( [I\bullet(iP_{2}),T]_{*}=0 \). So \( T_{12}=T_{21}=T_{22}=0 \). Similarly, we can show that \( T_{11}=0 \), proving Step 2.3.
Step 2.4. Given \( A_{jk},B_{jk}\in{\mathcal{A}}_{jk} \), with \( 1\leq j\neq k\leq 2 \), we have \( \Phi(A_{jk}+B_{jk})=\Phi(A_{jk})+\Phi(B_{jk}) \).
Since
we get from Step 2.3 that
Then \( \Phi(A_{jk}+B_{jk})=\Phi(A_{jk})+\Phi(B_{jk}) \).
Step 2.5. Given \( A_{jj},B_{jj}\in{\mathcal{A}}_{jj} \), with \( 1\leq j\leq 2 \), we have
Let \( T=\Phi(A_{jj}+B_{jj})-\Phi(A_{jj})-\Phi(B_{jj}) \). For \( 1\leq j\neq k\leq 2 \), it follows that
From this we get \( [I\bullet(iP_{k}),T]_{*}=0 \). So \( T_{jk}=T_{kj}=T_{kk}=0 \). Now \( T=T_{jj} \).
For all \( C_{jk}\in{\mathcal{A}}_{jk} \), \( j\neq k \), it follows from Step 2.4 that
Hence \( [I\bullet T_{jj},C_{jk}]_{*}=0 \) for all \( C_{jk}\in{\mathcal{A}}_{jk} \); i.e., \( T_{jj}CP_{k}=0 \) for all \( C\in{\mathcal{A}} \). It follows from \( (\spadesuit) \) and \( (\clubsuit) \) that \( T=T_{jj}=0 \), proving the step.
Now, it follows from Steps 2.3–2.5 that \( \Phi \) is additive, proving Claim 2.
Claim 3
\( \Phi(I) \) is a self-adjoint central element in \( \mathcal{A} \).
On the one hand,
which implies that \( \Phi(I) \) is a self-adjoint element in \( {\mathcal{A}} \).
On the other hand, for all \( A\in{\mathcal{A}} \) we get
which implies that \( \Phi(I) \) is a central element in \( {\mathcal{A}} \).
Claim 4
\( P_{1}\Phi(P_{1})P_{2}=-P_{1}\Phi(P_{2})P_{2} \), \( P_{2}\Phi(P_{1})P_{1}=-P_{2}\Phi(P_{2})P_{1} \), and \( P_{1}\Phi(P_{2})P_{1}=P_{2}\Phi(P_{1})P_{2}=0 \).
On the one hand, for \( 1\leq j\neq k\leq 2 \), it follows from Claim 3 that
Multiplying both sides of the above equation by \( P_{j} \) and \( P_{k} \) from the left and right, respectively, we infer that \( P_{1}\Phi(P_{1})P_{2}=-P_{1}\Phi(P_{2})P_{2} \) and \( P_{2}\Phi(P_{1})P_{1}=-P_{2}\Phi(P_{2})P_{1} \).
On the other hand, we get
Multiplying both sides of the above equation by \( P_{j} \) from the left and right, respectively, we obtain that \( P_{1}\Phi(P_{2})P_{1}=P_{2}\Phi(P_{1})P_{2}=0 \).
Claim 5
\( P_{1}\Phi(P_{1})P_{1}=P_{2}\Phi(P_{2})P_{2}=0 \).
For every \( A_{12}\in{\mathcal{A}}_{12} \), on the one hand, it follows from Claims 2 and 3 that
Multiplying both sides of the above equation by \( P_{1} \) and \( P_{2} \) from the left and right, respectively, by Claim 4, we get that
On the other hand, we have
Multiplying both sides of the above equation by \( P_{1} \) and \( P_{2} \) from the left and right, respectively, we get that
Finally,
Multiplying both sides of the above equation by \( P_{1} \) and \( P_{2} \) from the left and right, respectively, by Claim 4, we get that
It follows from (2.2) and (2.3) that
Now, by (2.1) and (2.4), we have \( P_{1}\Phi(P_{1})A_{12}=0 \); i.e., \( P_{1}\Phi(P_{1})P_{1}AP_{2}=0 \) for all \( A\in{\mathcal{A}} \). It follows from \( (\clubsuit) \) that \( P_{1}\Phi(P_{1})P_{1}=0 \). Similarly, we can prove that \( P_{2}\Phi(P_{2})P_{2}=0 \).
Claim 6
\( \Phi(I)=0 \).
By Claims 2, 4, and 5, we can get that
Claim 7
\( \Phi([A,B]_{*})=[\Phi(A),B]_{*}+[A,\Phi(B)]_{*} \) for all \( A,B\in{\mathcal{A}} \).
It follows from Claims 2 and 6 that
which implies that \( \Phi([A,B]_{*})=[\Phi(A),B]_{*}+[A,\Phi(B)]_{*} \).
Claim 8
\( \Phi(A^{*})=\Phi(A)^{*} \) for all \( A\in{\mathcal{A}} \).
For every \( A\in{\mathcal{A}} \), by Claims 2, 6, and 7, we have
Hence \( \Phi(A^{*})=\Phi(A)^{*} \).
Claim 9
\( \Phi(iI)=0 \).
By Claims 2 and 8, we can get \( \Phi(iI)^{*}=-\Phi(iI) \). So
which implies that \( \Phi(iI)=0 \).
Claim 10
\( \Phi(iA)=i\Phi(A) \) for all \( A\in{\mathcal{A}} \).
It follows from Claims 2 and 9 that
and then \( \Phi(iA)=i\Phi(A) \).
Claim 11
\( \Phi \) is a derivation.
On the one hand, by Claim 7, we have
On the other hand, by Claims 2, 7, and 10, we also have
From (2.5) and (2.6) we obtain \( \Phi(AB)=\Phi(A)B+A\Phi(B) \).
Now, by Claims 2, 8, and 11, we have proved that \( \Phi \) is an additive \( * \)-derivation. This completes the proof of Theorem 2.1. ☐
3. Corollaries
In this section, we present some corollaries of the main result. An algebra \( {\mathcal{A}} \) is called prime if \( A\mathcal{A}B=\{0\} \) for \( A,B\in{\mathcal{A}} \) implies either \( A=0 \) or \( B=0 \). It is easy to see that prime \( * \)-algebras satisfy \( (\spadesuit) \) and \( (\clubsuit) \). So we have the following corollary.
Corollary 3.1
Let \( {\mathcal{A}} \) be a prime \( * \)-algebra with unit \( I \) and let \( P \) be a nontrivial projection in \( {\mathcal{A}} \). Then \( \Phi \) is a nonlinear mixed Jordan triple \( * \)-derivation on \( {\mathcal{A}} \) if and only if \( \Phi \) is an additive \( * \)-derivation.
We denote by \( B({\mathcal{H}}) \) the algebra of all bounded linear operators on a complex Hilbert space \( {\mathcal{H}} \) and by \( {\mathcal{F}}(H)\subseteq B({\mathcal{H}}) \), the subalgebra of all bounded finite rank operators. A subalgebra \( {\mathcal{A}}\subseteq B({\mathcal{H}}) \) is called a standard operator algebra if \( {\mathcal{A}} \) includes \( {\mathcal{F}}(H) \). Now we have the following corollary.
Corollary 3.2
Let \( {\mathcal{A}} \) be a standard operator algebra on an infinite-dimensional complex Hilbert space \( {\mathcal{H}} \) containing the identity operator \( I \). Suppose that \( {\mathcal{A}} \) is closed under the adjoint operation. Then \( \Phi:{\mathcal{A}}\rightarrow{\mathcal{A}} \) is a nonlinear mixed Jordan triple \( * \)-derivation if and only if \( \Phi \) is a linear \( * \)-derivation. Moreover, there exists an operator \( T\in B({\mathcal{H}}) \) satisfying \( T+T^{*}=0 \) such that \( \Phi(A)=AT-TA \) for all \( A\in A \), i.e., \( \Phi \) is inner.
Proof
Since \( {\mathcal{A}} \) is prime, we have that \( \Phi \) is an additive \( * \)-derivation. It follows from [28] that \( \Phi \) is a linear inner derivation, i.e., there exists an operator \( S\in B({\mathcal{H}}) \) such that \( \Phi(A)=AS-SA \). Since \( \Phi(A^{*})=\Phi(A)^{*} \), we have
for all \( A\in A \). Hence
and then \( S+S^{*}=\lambda I \) for some \( \lambda\in{} \). Let
It is easy to see that \( T+T^{*}=0 \) such that \( \Phi(A)=AT-TA \). ☐
A von Neumann algebra \( {\mathcal{M}} \) is a weakly closed self-adjoint algebra of operators on a Hilbert space \( {\mathcal{H}} \) containing the identity operator \( I \). Note that \( {\mathcal{M}} \) is a factor von Neumann algebra if its center only contains the scalar operators. It is well known that a factor von Neumann algebra is prime. So we have the following corollary:
Corollary 3.3
Let \( {\mathcal{M}} \) be a factor von Neumann algebra with \( \dim{\mathcal{M}}\geq 2 \). Then \( \Phi:{\mathcal{M}}\rightarrow{\mathcal{M}} \) is a nonlinear mixed Jordan triple \( * \)-derivation if and only if \( \Phi \) is an additive \( * \)-derivation.
It is shown in [2] and [18] that if a von Neumann algebra has no central summands of type \( I_{1} \), then \( {\mathcal{M}} \) satisfies \( (\spadesuit) \) and \( (\clubsuit) \). Now we have the following corollary:
Corollary 3.4
Let \( {\mathcal{M}} \) be a von Neumann algebra with no central summands of type \( I_{1} \). Then \( \Phi:{\mathcal{M}}\rightarrow{\mathcal{M}} \) is a nonlinear mixed Jordan triple \( * \)-derivation if and only if \( \Phi \) is an additive \( * \)-derivation.
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The authors thank the referee for the very thorough reading of the paper and many helpful comments.
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Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 4, pp. 884–892. https://doi.org/10.33048/smzh.2022.63.414
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Li, C., Zhang, D. Nonlinear Mixed Jordan Triple \( * \)-Derivations on \( * \)-Algebras. Sib Math J 63, 735–742 (2022). https://doi.org/10.1134/S0037446622040140
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DOI: https://doi.org/10.1134/S0037446622040140