In this paper, the authors introduce the concept of a Lie triple centralizer on generalized matrix algebras. A linear map \(\varphi: A \to A\) is defined as a Lie triple centralizer if it satisfies \(\varphi([[a,b],c]) = [[\varphi(a),b],c]\) for all \(a, b, c \in A\). Below are the key theorems presented in the paper:
Definitions of symbols:

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\(U\) denotes a generalized matrix algebra, defined as \(U = \begin{bmatrix} A & M \\ N & B \end{bmatrix}\), where \(A\) and \(B\) are algebras, \(M\) is an \((A,B)\)-bimodule, and \(N\) is a \((B,A)\)-bimodule.

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\(Z(A)\) denotes the center of an algebra \(A\), defined as \(Z(A) = \{a \in A : ac = ca \text{ for all } c \in A\}\).

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\(Z(U)\) denotes the center of the algebra \(U\), with \(Z(U) = \begin{bmatrix} Z(A) & 0 \\ 0 & Z(B) \end{bmatrix}\) under certain conditions.

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\(M\) and \(N\) are bimodules that satisfy the conditions: if \(a \in A\), \(aM = 0\), and \(Na = 0\), then \(a = 0\). Similarly, if \(b \in B\), \(Mb = 0\), and \(bN = 0\), then \(b = 0\).

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\(\pi_A\) and \(\pi_B\) are projection mappings defined as \(\pi_A\left(\begin{bmatrix} a & m \\ n & b \end{bmatrix}\right) = a\) and \(\pi_B\left(\begin{bmatrix} a & m \\ n & b \end{bmatrix}\right) = b\).

Main theorems:

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Theorem 3.1: A linear map \(\varphi: U \to U\) is a Lie triple centralizer if and only if \(\varphi\) has the form: \[ \varphi \left( \begin{bmatrix} a & m \\ n & b \end{bmatrix} \right) = \begin{bmatrix} \alpha_1(a) + \beta_1(b) & \tau_2(m) \\ \gamma_3(n) & \alpha_4(a) + \beta_4(b) \end{bmatrix} \] where \(\alpha_1 : A \to A\), \(\beta_1 : B \to [A,A]'\), \(\tau_2 : M \to M\), \(\gamma_3 : N \to N\), \(\alpha_4 : A \to [B,B]'\), and \(\beta_4 : B \to B\) are linear mappings satisfying the following conditions:

(1)

\(\alpha_1\) is a Lie triple centralizer on \(A\), \(\alpha_4([[a_1, a_2], a_3]) = 0\), and \(\alpha_1(mn) - \beta_1(nm) = \tau_2(m)n = m\gamma_3(n)\) for all \(m \in M\) and \(n \in N\).

(2)

\(\beta_4\) is a Lie triple centralizer on \(B\), \(\beta_1([[b_1, b_2], b_3]) = 0\), and \(\alpha_4(nm) - \beta_4(mn) = n\tau_2(m) = \gamma_3(n)m\) for all \(m \in M\) and \(n \in N\).

(3)

\(\tau_2(am) = a\tau_2(m) = \alpha_1(a)m - m\alpha_4(a)\) and \(\tau_2(mb) = \tau_2(m)b = m\beta_4(b) - \beta_1(b)m\) for any \(a \in A\), \(b \in B\), and \(m \in M\).

(4)

\(\gamma_3(na) = \gamma_3(n)a = n\alpha_1(a) - \alpha_4(a)n\) and \(\gamma_3(bn) = b\gamma_3(n) = \beta_4(b)n - n\beta_1(b)\) for any \(a \in A\), \(b \in B\), and \(n \in N\).

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Theorem 3.3: Let \(U\) satisfy the conditions: \begin{align*} & a \in A, aM = 0 \text{ and } Na = 0 \Rightarrow a = 0, \\ & b \in B, Mb = 0 \text{ and } bN = 0 \Rightarrow b = 0. \end{align*} A linear map \(\varphi : U \to U\) is a proper Lie triple centralizer if and only if \(\alpha_4(A) \subseteq \pi_B(Z(U))\) and \(\beta_1(B) \subseteq \pi_A(Z(U))\).

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Corollary 3.4: Under the same conditions as Theorem 3.3, if either \(\pi_B(Z(U)) = Z(B)\) or \([[A,A],A] = A\) holds, then any Lie triple centralizer on \(U\) is a proper Lie triple centralizer.

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Theorem 4.2: Let \(U\) satisfy the conditions stated in Theorem 3.3. If \(\Lambda: U \to U\) is a generalized Lie triple derivation associated with a Lie triple derivation \(\xi: U \to U\), then \(\Lambda(A) = \delta(A) + d(A) + \psi(A) + \lambda A\) for any \(A \in U\), where \(\delta\) is a derivation on \(U\), \(d\) is a singular Jordan derivation on \(U\), \(\lambda \in Z(U)\), and \(\psi: U \to Z(U)\) is a linear map that vanishes on \([[U, U], U]\).

These results extend the theory of Lie triple centralizers and provide a foundation for further studies in the Lie structure of algebras.