EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5626 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Note on Nonlinear Mixed (bi-Skew, skew Lie) Triple Derivations on ∗-Algebras M. Arif Raza1, Junaid Nisar2,∗, Nadeem ur Rehman3, Vahid Darvish4 1 Department of Mathematics, Faculty of Science & Arts-Rabigh, King Abdulaziz University, KSA 2 Department of Applied Sciences, Symbiosis Institute of Technology, Symbiosis International (Deemed) University, Lavale, Pune,India 3 Department of Mathematics, Aligarh Muslim University, Aligarh-202002 India 4 School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China Abstract. Let A be a unital ∗-algebra containing non-trivial projection. We prove that if a map Λ : A → A such that Λ([[L,M]•,N]∗) = [[Λ(L),M]•,N]∗ + [[L,Λ(M)]•,N]∗ + [[L,M]•,Λ(N)]∗ for all L,M,N ∈ A, then Λ is additive. Moreover, if Λ(I) is self-adjoint, then Λ is a ∗-derivation. Additionally, as an application, we can also apply our results on factor von Neumann algebras, standard operator algebras and prime ∗-algebras. 2020 Mathematics Subject Classifications: 16W10, 47B47, 46K15. Key Words and Phrases: Mixed bi-skew Lie triple derivation, ∗-derivation, ∗- algebra 1. Introduction Let A be an ∗-algebra over the complex field C. For L,M ∈ A, we call [L,M]∗ = LM − ML∗ the skew Lie product and [L,M]• = LM∗ − ML∗ denotes the bi-skew Lie product. The skew Lie product, Jordan product, and bi-skew Lie product have become increasingly relevant in various research fields, and numerous authors have shown a keen interest in their exploration. This is evident from the numerous studies by authors (see [1, 2, 4–7, 9, 10, 13]). Recall that an additive map Λ : A → A is called an additive derivation if Λ(LM) = Λ(L)M+LΛ(M) for all L,M ∈ A. If Λ(L∗) = Λ(L)∗ for all L ∈ A, then Λ is an additive ∗-derivation. Let Λ : A → A be a map (without the additivity assumption). We say Λ is a nonlinear skew Lie derivation or nonlinear skew Lie triple derivation if Λ([L,M]∗) = [Λ(L),M]∗ + [L,Λ(M)]∗ ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5626 Email addresses: arifraza03@gmail.com (M. Raza), junaidnisar73@gmail.com (J. Nisar), nu.rehman.mm@amu.ac.in (N. Rehman), vdarvish@nuist.edu.cn (V. Darvish) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. A. Raza et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5626 2 of 10 or Λ([[L,M]∗,N]∗) = [[Λ(L),M]∗,N]∗ + [[L,Λ(M)]∗,N]∗ + [[L,M]∗,Λ(N)]∗ for all L,M,N ∈ A. Similarly, a map Λ : A → A is said to be a nonlinear bi-skew Lie derivation or nonlinear bi-skew Lie triple derivation if Λ([L,M]•) = [Λ(L),M]• + [L,Λ(M)]• or Λ([[L,M]•,N]•) = [[Λ(L),M]•,N]• + [[L,Λ(M)]•,N]∗ + [[L,M]•,Λ(N)]• for all L,M,N ∈ A. In 2021, A. Khan [3] established a proof demonstrating that any mul- tiplicative or nonadditive bi-skew Lie triple derivation acting on a factor Von Neumann algebra can be characterized as an additive ∗-derivation. Numerous authors have recently explored the derivations and isomorphisms correspond- ing to the novel products created by combining Lie and skew Lie products, skew Lie and skew Jordan product see [8, 11, 12]. As an illustration, Li and Zhang [8] delved into an investigation focused on understanding the arrangement and properties of the nonlinear mixed Jordan triple ∗-derivation within the domain of ∗-algebras. In 2023, Rehman et. al. [12] mixed the concept of Jordan and Jordan ∗-product and gives the complete character- ization of nonlinear mixed Jordan ∗-triple derivation on ∗-algebras. Inspired by the above results, in the present paper, we combined skew Lie product and bi-skew Lie product and defined nonlinear mixed bi-skew Lie triple derivations on ∗-algebras. A map Λ: A → A is called nonlinear mixed bi-skew Lie triple derivations if Λ([[L,M]•,N]∗) = [[Λ(L),M]•,N]∗ + [[L,Λ(M)]•,N]∗ + [[L,M]•,Λ(N)]∗ for all L,M,N ∈ A.Our proof establishes that when Λ represents a nonlinear mixed bi-skew Lie triple derivation acting on ∗-algebras, it necessarily possesses additivity. Furthermore, if the image of Λ under the transformation of the identity element (Λ(I)) is self-adjoint, then Λ can be identified as an ∗-derivation. In simpler terms, the study demonstrates that specific properties, such as additivity and self-adjointness, can be attributed to the nature of nonlinear mixed bi-skew Lie triple derivations on ∗-algebras. 2. Main Result Our First Theorem is as follows: Theorem 2.1. Let A be a unital ∗-algebra with unity I containing a non-trivial projection P satisfies XAP = 0 =⇒ X = 0 (▲) and XA(I− P) = 0 =⇒ X = 0. (▼) M. A. Raza et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5626 3 of 10 Define a map Λ : A → A such that Λ([[L,M]•,N]∗) = [[Λ(L),M]•,N]∗ + [[L,Λ(M)]•,N]∗ + [[L,M]•,Λ(N)]∗ Then Λ is an additive. Proof. Let P = P1 be a non-trivial projection in A and P2 = I − P1, where I is the unity of this algebra. Then by Peirce decomposition of A, we have A = P1AP1⊕P1AP2⊕ P2AP1⊕P2AP2 and, denote A11 = P1AP1,A12 = P1AP2,A21 = P2AP1 and A22 = P2AP2. Note that any L ∈ A can be written as L = L11 + L12 + L21 + L22, where Lij ∈ Aij and L∗ ij ∈ Aji for i, j = 1, 2. Several lemmas are used to prove Theorem 2.1. Lemma 2.1. Λ(0) = 0. Proof. It is trivial that Λ(0) = Λ([[0, 0]•, 0]∗) = [[Λ(0), 0]•, 0]∗ + [[0,Λ(0)]•, 0]∗ + [[0, 0]•,Λ(0)]∗ = 0. Lemma 2.2. For any Lij ∈ Aij , 1 ≤ i, j ≤ 2, we have Λ( 2∑ i,j=1 Lij) = 2∑ i,j=1 Λ(Lij). Proof. Let M = Λ(L11 + L12 + L21 + L22) − Λ(L11) − Λ(L12) − Λ(L21) − Λ(L22). In order to prove that Λ(L11 + L12 + L21 + L22) = Λ(L11) + Λ(L12) + Λ(L21) + Λ(L22), we show M = 0. Since [[L12,P1]•,P1]∗ = [[L21,P1]•,P1]∗ = [[L22,P1]•,P1]∗ = 0. It follows from Lemma 2.1 that Λ([[L11 + L12 + L21 + L22,P1]•,P1]∗) = Λ([[L11,P1]•,P1]∗) + Λ([[L12,P1]•,P1]∗) +Λ([[L21,P1]•,P1]∗) + Λ([[L22,P1]•,P1]∗) = [[Λ(L11) + Λ(L12) + Λ(L21) + Λ(L22),P1]•,P1]∗ +[[L11 + L12 + L21 + L22,Λ(P1)]•,P1]∗ +[[L11 + L12 + L21 + L22,P1]•,Λ(P1)]∗ and Λ([[L11 + L12 + L21 + L22,P1]•,P1]∗) = [[Λ(L11 + L12 + L21 + L22),P1]•,P1]∗ +[[L11 + L12 + L21 + L22,Λ(P1)]•,P1]∗ +[[L11 + L12 + L21 + L22,P1]•,Λ(P1)]∗. From the above equations, we get [[M,P1]•,P1]∗ = 0. This implies that MP1−P1M ∗P1− P1M ∗ + P1MP1 = 0. By multiplying P2 from left, we get P2MP1 = 0. Similarly, by M. A. Raza et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5626 4 of 10 applying P2 instead of P1, we get P1MP2 = 0. Also, for any X12 ∈ A12, we have Λ([[L11 + L12 + L21 + L22,X12]•,P2]∗) = [[Λ(L11 + L12 + L21 + L22),X12]•,P2]∗ +[[L11 + L12 + L21 + L22,Λ(X12)]•,P2]∗ +[[L11 + L12 + L21 + L22,X12]•,Λ(P2)]∗. From Lemma 2.1, we get Λ([[L11 + L12 + L21 + L22,X12]•,P2]∗) = Λ([[L11,X12]•,P2]∗) + Λ([[L12,X12]•,P2]∗) +Λ([[L21,X12]•,P2]∗) + Λ([[L22,X12]•,P2]∗) = [[Λ(L11),X12]•,P2]∗ + [[L11,Λ(X12)]•,P2]∗ +[[L11,X12]•,Λ(P2)]∗ + [[Λ(L12),X12]•,P2]∗ +[[L12,Λ(X12)]•,P2]∗ + [[L12,X12]•,Λ(P2)]∗ +[[Λ(L21),X12]•,P2]∗ + [[L21,Λ(X12)]•,P2]∗ +[[L21,X12]•,Λ(P2)]∗ + [[Λ(L22),X12]•,P2]∗ +[[L22,Λ(X12)]•,P2]∗ + [[L22,X12]•,Λ(P2)]∗. From the above two equations, we get [[M,X12]•,P2]∗ = 0. That means −X12M ∗P2 + P2MX∗ 12 = 0. By multiplying P1 from left, we get P2MX∗ 12 = 0. Thus, P2MP2 = 0 by using (▲) and (▼). In the similar way, we can show that P1MP1 = 0 by choosing X21 and P1 instead of X21 and P1 respectively in above. Hence M = 0. It follows that Λ( ∑2 i,j=1Lij) = ∑2 i,j=1 Λ(Lij). Lemma 2.3. For each L12,M12 ∈ A12 and L21,M21 ∈ A21, we have (i) Λ(L12 +M12) = Λ(L12) + Λ(M12). (ii) Λ(L21 +M21) = Λ(L21) + Λ(M21). Proof. (1) Let T = Λ(L12 +M12)−Λ(L12)−Λ(M12). It follows from Lemma 2.1 that Λ([[L12 +M12,P1]•,P2]∗) = Λ([[L12,P1]•,P2]∗) + Λ([[M12,P1]•,P2]∗) = [[Λ(L12),P1]•,P2]∗ + [[L12,Λ(P1)]•,P2]∗ + [[L12,P1]•,Λ(P2)]∗ +[[Λ(M12),P1]•,P2]∗ + [[M12,Λ(P1)]•,P2]∗ + [[M12,P1]•,Λ(P2)]∗. Alternatively, we have Λ([[L12 +M12,P1]•,P2]∗) = [[Λ(L12 +M12),P1]•,P2]∗ + [[L12 +M12,Λ(P1)]•,P2]∗ +[[L12 +M12,P1]•,Λ(P2)]∗. By comparing the above two expressions, we get [[T,P1]•,P2]∗ = 0. This implies that P2TP1 = 0. Similarly, P1TP2 = 0. For any X12 ∈ A12, we have Λ([[X12,L12 +M12]•,P2]∗) = [[Λ(X12),L12 +M12]•,P2]∗ + [[X12,Λ(L12 +M12)]•,P2]∗ M. A. Raza et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5626 5 of 10 +[[X12,L12 +M12]•,Λ(P2)]∗. Since [[X12,L12]•,P2]∗ = 0 and using Lemma 2.1, we have Λ([[X12,L12 +M12]•,P2]∗) = Λ([[X12,L12]•,P2]∗) + Λ([[X12,M12]•,P2]∗) = [[Λ(X12),L12]•,P2]∗ + [[X12,Λ(L12)]•,P2]∗ +[[X12,L12]•,Λ(P2)]∗ + [[Λ(X12),M12]•,P2]∗ +[[X12,Λ(M12)]•,P2]∗ + [[X12,M12]•,Λ(P2)]∗. From the last two expressions, we get [[X12, T ]•,P2]∗ = 0. That means X12T ∗P2 − P2MX∗ 12 = 0. Multiplying left side by P2 and then using (▲) and (▼), we get P2TP2 = 0. Similarly, P1TP1 = 0. Hence, T = 0. (2) By using the similar argument as in (1), we get the required conclusion. Lemma 2.4. For each Lii,Mii ∈ Aii such that 1 ≤ i ≤ 2, we have Λ(Lii +Mii) = Λ(Lii) + Λ(Mii). Proof. Let T = Λ(Lii +Mii)− Λ(Lii)− Λ(Mii). It follows from Lemma 2.1 and i ̸= j that Λ([[Pj ,Lii +Mii]•,Pi]∗) = Λ([[Pj ,Lii]•,Pi]∗) + Λ([[Pj ,Mii]•,Pi]∗) = [[Λ(Pj),Lii]•,Pi]∗ + [[Pj ,Λ(Lii)]•,Pi]∗ + [[Pj ,Lii]•,Λ(Pi)]∗ +[[Λ(Pj),Mii]•,Pi]∗ + [[Pj ,Λ(Mii)]•,Pi]∗ + [[Pj ,Mii]•,Λ(Pi)]∗ and Λ([[Pj ,Lii +Mii]•,Pi]∗) = [[Λ(Pj),Lii +Mii]•,Pi]∗ + [[Pj ,Λ(Lii +Mii)]•,Pi]∗ +[[Pj ,Lii +Mii]•,Λ(Pi)]∗. By comparing the last two expressions, we get [[Pj , T ]•,Pi]∗ = 0. This gives PiTPj = 0 with i ̸= j. Also, for any Xij ∈ Aij , we have Λ([[Xij ,Lii +Mii]•,Pi]∗) = [[Λ(Xij),Lii +Mii]•,Pi]∗ + [[Xij ,Λ(Lii +Mii)]•,Pi]∗) +[[Xij ,Lii +Mii]•,Λ(Pi)]∗. Under other conditions, [[Xij ,Lii]•,Pi]∗ = 0 and using Lemma 2.1, we have Λ([[Xij ,Lii +Mii]•,Pi]∗) = Λ([[Xij ,Lii]•,Pi]∗) + Λ([[Xij ,Mii]•,Pi]∗) = [[Λ(Xij),Lii]•,Pi]∗ + [[Xij ,Λ(Lii)]•,Pi]∗ + [[Xij ,Lii]•,Λ(Pi)]∗ +[[Λ(Xij),Mii]•,Pi]∗ + [[Xij ,Λ(Mii)]•,Pi]∗ + [[Xij ,Mii]•,Λ(Pi)]∗. From the last two expressions, we get [[Xij , T ]•,Pi]∗ = 0. That means XijT ∗Pi − TX∗ ij − PiTX ∗ ij + XijT ∗ = 0. Left multiplying by Pj both sides and using (▲) and (▼), we find PjTPj = 0. M. A. Raza et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5626 6 of 10 Lemma 2.5. Λ is an additive map. Proof. For any L,M ∈ A, we write L = L11 +L12 +L21 +L22 and M = M11 +M12 + M21 +M22. By using Lemmas 2.2 - 2.4, we get Λ(L+M) = Λ(L11 + L12 + L21 + L22 +M11 +M12 +M21 +M22) = Λ(L11 +M11) + Λ(L12 +M12) + Λ(L21 +M21) + Λ(L22 +M22) = Λ(L11) + Λ(M11) + Λ(L12) + Λ(M12) + Λ(L21) + Λ(M21) + Λ(L22) + Λ(M22) = Λ(L11 + L12 + L21 + L22) + Λ(M11 +M12 +M21 +M22) = Λ(L) + Λ(M). Hence, Λ is additive. This completes the proof of Theorem 2.1. Theorem 2.2. Let A be a unital ∗-algebra with unity I containing a non-trivial projection P satisfies (▲) and (▼). Let the map Λ : A → A satisfy the condition Λ([[L,M]•,N]∗) = [[Λ(L),M]•,N]∗ + [[L,Λ(M)]•,N]∗ + [[L,M]•,Λ(N)]∗ for L,M,N ∈ A. If Λ(I) is self-adjoint, then Λ is an ∗-derivation. Proof of Theorem 2.2 We present the proof of the above theorem with several lemmas. Lemma 2.6. We show that if Λ(I) is self-adjoint then Λ(iI) = Λ(I) = 0. Proof. we know that Λ([[iI, I]•, I]∗) = [[Λ(iI), I]•, I]∗ + [[iI,Λ(I)]•, I]∗ + [[iI, I]•,Λ(I)]∗ = 2Λ(iI)− 2Λ(iI)∗ + 2iΛ(I)∗ + 2iΛ(I) + 4iΛ(I). Also, from the other side, we have Λ([[iI, I]•, I]∗) = 4Λ(iI). By using above two equations, we get 2Λ(iI)− 2Λ(iI)∗ + 2iΛ(I)∗ + 2iΛ(I) + 4iΛ(I)− 4Λ(iI) = 0. (2.1) Alternatively, we have Λ([[iI, I]•, iI]∗) = −4Λ(I). Also, we have Λ([[iI, I]•, iI]∗) = 2iΛ(iI)− 2iΛ(iI)∗ − 2Λ(I)∗ − 2Λ(I) + 4iΛ(iI). M. A. Raza et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5626 7 of 10 From the last two expressions, we have 4Λ(I) + 2iΛ(iI)− 2iΛ(iI)∗ − 2Λ(I)∗ − 2Λ(I) + 4iΛ(iI) = 0 (2.2) Multiplying (2.2) by i, we get 4iΛ(I)− 2Λ(iI) + 2Λ(iI)∗ − 2iΛ(I)∗ − 2iΛ(I)− 4Λ(iI) = 0 (2.3) Adding (2.1) and (2.3), we get Λ(iI) = iΛ(I). (2.4) Using (2.4) in (2.3), we get Λ(I)∗ = −Λ(I). (2.5) Since Λ(I) is self-adjoint, then Λ(I) = Λ(iI) = 0. Lemma 2.7. Λ preserves star, i.e., Λ(L∗) = Λ(L)∗ for all L ∈ A. Proof. From Lemma 2.6, we have Λ([[L, iI]•, iI]∗) = [[Λ(L), iI]•, iI]∗ = [[−iΛ(L)− iΛ(L)∗, iI]∗ = 2Λ(L) + 2Λ(L)∗. On the other hand, we have Λ([[L, iI]•, iI]∗) = 2Λ(L) + 2Λ(L∗). From the last two equations, we get Λ(L∗) = Λ(L)∗. Lemma 2.8. We prove that Λ(iL) = iΛ(L) for all L ∈ A. Proof. It follows from Lemma 2.6 that Λ([[iL, I]•, I]∗) = [Λ(iL), I]•, I]∗ = 2Λ(iL)− 2Λ(iL)∗. Hence Λ(2iL+ 2iL∗) = 2Λ(iL)− 2Λ(iL)∗. (2.6) From the other side, we have Λ([[L, iI]•, I]∗) = [Λ(L), iI]•, I]∗ = −2iΛ(L)− 2iΛ(L)∗ It follows that Λ(−2iL− 2iL∗) = −2iΛ(L)− 2iΛ(L)∗. (2.7) M. A. Raza et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5626 8 of 10 Adding (2.6) and (2.7), we get Λ(i(L+ L∗)) = iΛ(L+ L∗). (2.8) Since (2.8) is true for any self-adjoint then for any member of L, we have Λ(iL) = iΛ(L). Lemma 2.9. We show that Λ is a derivation, i.e, Λ(LM) = Λ(L)M+ LΛ(M). Proof. It is easy to check that Λ([[L,M]•, I]∗) = 2Λ(LM∗)− 2Λ(ML∗). Also, it follows from Lemma 2.6 that Λ([[L,M]•, I]∗) = [[Λ(L),M]•, I]∗ + [[L,Λ(M)]•, I]∗ = 2Λ(L)M∗ − 2MΛ(L)∗ + 2LΛ(M)∗ − 2Λ(M)L∗. By comparing the last two expressions, we have Λ(LM∗)− Λ(ML∗) = Λ(L)M∗ −MΛ(L)∗ + LΛ(M)∗ − Λ(M)L∗ (2.9) On the other hand, we have Λ([[iL,M]•, iI]∗) = −Λ(LM∗)− Λ(ML∗). By using Lemma 2.6 and Lemma 2.8, we have Λ([[iL,M]•, iI]∗) = [[Λ(iL),M]•, iI]∗ + [[iL,Λ(M)]•, iI]∗ = iΛ(iL)M∗ − iMΛ(iL)∗ − LΛ(M)∗ − Λ(M)L∗ = −Λ(L)M∗ −MΛ(L)∗ − LΛ(M)∗ − Λ(M)L∗. By comparing the last two expressions, we have Λ(LM∗) + Λ(ML∗) = Λ(L)M∗ +MΛ(L)∗ + LΛ(M)∗ + Λ(M)L∗ (2.10) Adding (2.9) and (2.10), we get Λ(LM∗) = Λ(L)M∗ + LΛ(M∗). (2.11) Replacing M∗ by M, we get Λ(LM) = Λ(L)M+ LΛ(M). Hence, Λ is a derivation. This completes the proof of Theorem 2.2. Now, we provide an example to demonstrate the necessity of the conditions (▲) and (▼) in Theorem 2.1. M. A. Raza et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5626 9 of 10 Example 2.1. Consider A = { ( a 0 c d ) } , the algebra of all lower triangular matrix of order 2 over the field of complex numbers C and I = ( 1 0 0 1 ) be unity of A. The map ∗ : A → A given by ∗(L) = Lθ, where Lθ denotes the conjugate transpose of matrix A, is an involution. Hence, A is a unital ∗-algebra with unity I. Now, define a map Π : A → A such that Π ( a 0 c d ) = ( 0 0 −ic 0 ) . Note that Π is a derivation on A. So, it also satisfies Λ([[L,M]•,N]∗) = [[Λ(L),M]•,N]∗ + [[L,Λ(M)]•,N]∗ + [[L,M]•,Λ(N)]∗ for all L,M,N ∈ A. Let P = ( 0 0 0 1 ) is a non-trivial projection, so P 2 = P and P ∗ = P . For W = ( 0 0 1 0 ) ̸= 0 ∈ A and hence WAP = (0) but 0 ̸= W ∈ A. However, Π is not an additive ∗-derivation because Π(L∗) ̸= (Π(L))∗ for some L ∈ A. 3. Corollaries As a direct consequence of Theorem 2.1, we have the following corollaries: Corollary 3.1. Let A be a standard operator algebra on an infinite dimensional complex Hilbert space H containing identity operator I. Suppose that A is closed under adjoint operation. Define Λ : A → A such that Λ([[L,M]•,N]∗) = [[Λ(L),M]•,N]∗ + [[L,Λ(M)]•,N]∗ + [[L,M]•,Λ(N)]∗ for all L,M,N ∈ A, then Λ is an additive. If Λ(I) is self-adjoint, then Λ is an ∗-derivation. Corollary 3.2. Let M ba a factor von Neumann algebra with dimM ≥ 2. Define Λ : M → M such that Λ([[L,M]•,N]∗) = [[Λ(L),M]•,N]∗ + [[L,Λ(M)]•,N]∗ + [[L,M]•,Λ(N)]∗ for all L,M,N ∈ A, then Λ is an additive. If Λ(I) is self-adjoint, then Λ is an ∗-derivation. Corollary 3.3. 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