Linear Generalized n-Derivations on C^∗-Algebras

Abstract

Let $n\ge 2$ be a fixed integer and $\mathcal{A}$ be a $C^{\ast }$-algebra. A permuting n-linear map $\mathcal{G}:{\mathcal{A}}^n\to \mathcal{A}$ is known to be symmetric generalized n-derivation if there exists a symmetric n-derivation $\mathfrak{D}:{\mathcal{A}}^n\to \mathcal{A}$ such that $\mathcal{G}\left({\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_i{\varsigma }_i^{\prime },\dots ,{\varsigma }_n\right)=\mathcal{G}\left({\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_i,\dots ,{\varsigma }_n\right){\varsigma }_i^{\prime }+{\varsigma }_i\mathfrak{D}({\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_i^{\prime },\dots ,{\varsigma }_n)$ holds ∀${\varsigma }_i,{\varsigma }_i^{\prime }\in \mathcal{A}$. In this paper, we investigate the structure of $C^{\ast }$-algebras involving generalized linear n-derivations. Moreover, we describe the forms of traces of linear n-derivations satisfying certain functional identity.

1. Introduction

A Banach algebra is a linear associate algebra which, as a vector space, is a Banach space with norm $\left|\right|.\left|\right|$ satisfying the multiplicative inequality; $\left|\right|\varsigma \epsilon \left|\right|\le \left|\right|\varsigma \left|\right|\left|\right|\epsilon \left|\right|$∀$\varsigma$ and $\epsilon$ in $\mathcal{A}$. “An involution on an algebra $\mathcal{A}$ is a linear map $\varsigma \mapsto {\varsigma }^{\ast }$ of $\mathcal{A}$ into itself such that the following conditions hold: $(i)$${(\varsigma \epsilon )}^{\ast }={\epsilon }^{\ast }{\varsigma }^{\ast }$, $(ii)$${({\varsigma }^{\ast })}^{\ast }=\varsigma$, and $(iii)$${(\varsigma +\lambda \epsilon )}^{\ast }={\varsigma }^{\ast }+\overline{\lambda }{\epsilon }^{\ast }$∀$\varsigma ,\epsilon \in \mathcal{A}$ and $\lambda \in \mathbb{C}$, the field of complex numbers, where $\overline{\lambda }$ is the conjugate of $\lambda$. An algebra equipped with an involution ∗ is called a ∗-algebra or algebra with involution. A Banach ∗-algebra is a Banach algebra $\mathcal{A}$ together with an isometric involution $\left|\right|{\varsigma }^{\ast }\left|\right|=\left|\right|\varsigma \left|\right|$∀$\varsigma \in \mathcal{A}$. A $C^{\ast }$-algebra $\mathcal{A}$ is a Banach ∗-algebra with the additional norm condition $\left|\right|{\varsigma }^{\ast }\varsigma \left|\right|={\left|\right|\varsigma \left|\right|}^2$∀$\varsigma \in \mathcal{A}$.

Throughout this discussion, unless otherwise mentioned, $\mathcal{A}$ will denote $C^{\ast }$-algebra with $\mathcal{Z}(\mathcal{A})$ as its center. However, $\mathcal{A}$ may or may not have unity. The symbols $[\varsigma ,\epsilon ]$ and $\varsigma \circ \epsilon$ denote the commutator $\varsigma \epsilon -\epsilon \varsigma$ and the anti-commutator $\varsigma \epsilon +\epsilon \varsigma$, respectively, for any $\varsigma ,\epsilon \in \mathcal{A}$. An algebra $\mathcal{A}$ is said to be prime if $\varsigma \mathcal{A}\epsilon =\left\{0\right\}$ implies that either $\varsigma =0$ or $\epsilon =0$, and semiprime if $\varsigma \mathcal{A}\varsigma =\left\{0\right\}$ implies that $\varsigma =0$, where $\varsigma ,\epsilon \in \mathcal{A}$. An additive subgroup U of $\mathcal{A}$ is said to be a Lie ideal of $\mathcal{A}$ if $[u,r]\in U$, ∀$u\in U$, $r\in \mathcal{A}$. U is called a square-closed Lie ideal of $\mathcal{A}$ if U is a Lie ideal and $u^2\in U$∀$u\in U$. A linear operator $\mathcal{D}$ on a $C^{\ast }$-algebra $\mathcal{A}$ is called a derivation if $\mathcal{D}(\varsigma \epsilon )=\mathcal{D}(\varsigma )\epsilon +\varsigma \mathcal{D}(\epsilon )$ holds ∀$\varsigma ,\epsilon \in \mathcal{A}$. Consider the inner derivation ${\delta }_a$ implemented by an element a in $\mathcal{A}$, which is defined as ${\delta }_a(\varsigma )=\varsigma a-a\varsigma$ for every $\varsigma$ in $\mathcal{A}$, as a typical example of a nonzero derivation in a noncommutative algebra.

In order to broaden the scope of derivation, Maksa [1] introduced the concept of symmetric bi-derivations. A bi-linear map $\mathfrak{D}:\mathcal{A}\times \mathcal{A}\to \mathcal{A}$ is said to be a bi-derivation if

$$\mathfrak{D}(\varsigma {\varsigma }^{\prime },\epsilon )=\mathfrak{D}(\varsigma ,\epsilon ){\varsigma }^{\prime }+\varsigma \mathfrak{D}({\varsigma }^{\prime },\epsilon )$$

$$\mathfrak{D}(\varsigma ,\epsilon {\epsilon }^{\prime })=\mathfrak{D}(\varsigma ,\epsilon ){\epsilon }^{\prime }+\epsilon \mathfrak{D}(\varsigma ,{\epsilon }^{\prime })$$

holds for any $\varsigma ,{\varsigma }^{\prime },\epsilon ,{\epsilon }^{\prime }\in \mathcal{A}$. The foregoing conditions are identical if $\mathfrak{D}$ is also a symmetric map, whereby if $\mathfrak{D}(\varsigma ,\epsilon )=\mathfrak{D}(\epsilon ,\varsigma )$ for every $\varsigma ,\epsilon \in \mathcal{A}$. In this case, $\mathfrak{D}$ is referred to as a symmetric bi-derivation of $\mathcal{A}$. Vukman [2] investigated symmetric bi-derivations in prime and semiprime rings. Argao and Yenigül ([3], Chapter 3) and Muthana [4] obtained the similar type of results on Lie ideals of ring R.

In this paper, we briefly discuss the various extensions of the notion of derivations on $C^{\ast }$-algebras. The most general and important one among them is the notion of symmetric linear generalized n-derivations on $C^{\ast }$-algebras. Suppose n is a fixed positive integer and ${\mathcal{A}}^n=\mathcal{A}\times \mathcal{A}\times \cdots \times \mathcal{A}$. A map $\mathfrak{D}:{\mathcal{A}}^n\to \mathcal{A}$ is said to be symmetric (permuting) if the relation $\mathfrak{D}({\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_n)=\mathfrak{D}({\varsigma }_{\pi (1)},{\varsigma }_{\pi (2)},\dots ,{\varsigma }_{\pi (n)})$ holds ∀ ${\varsigma }_i\in \mathcal{A}$, $1\le i\le n$ and for every permutation $\{\pi (1),\pi (2),\dots ,\pi (n)\}$. The concept of derivation and symmetric bi-derivation was generalized by Park [5] as follows: a n-linear map $\mathfrak{D}:{\mathcal{A}}^n\to \mathcal{A}$ is said to be a symmetric (permuting) linear n-derivation if $\mathfrak{D}$ is permuting and $\mathfrak{D}({\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_i{\varsigma }_i^{\prime },\dots ,{\varsigma }_n)=\mathfrak{D}({\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_n){\varsigma }_i^{\prime }+{\varsigma }_i\mathfrak{D}({\varsigma }_1^{\prime },{\varsigma }_2,\cdots ,{\varsigma }_n)$ hold ∀${\varsigma }_i,{\varsigma }_i^{\prime }\in \mathcal{A},i=1,2,\dots ,n$. A map $\mathcal{d}:\mathcal{A}\to \mathcal{A}$ defined by $\mathcal{d}(\varsigma )=\mathcal{D}(\varsigma ,\varsigma ,\dots ,\varsigma )$ is called the trace of $\mathfrak{D}$. If $\mathfrak{D}:{\mathcal{A}}^n\to \mathcal{A}$ is permuting and n-linear, then the trace $\mathcal{d}$ of $\mathfrak{D}$ satisfies the relation

$$\mathcal{d}(\varsigma +\epsilon )=\mathcal{d}(\varsigma )+\mathcal{d}(\epsilon )+{\displaystyle \sum _{l=1}^{n-1}}{}^n{C}_l{h}_l(\varsigma ;\epsilon )$$

∀$\varsigma ,\epsilon \in \mathcal{A}$, where ${}^n{C}_l=\left(\genfrac{}{}{0pt}{}{n}{l}\right)$ and

$$h_l(\varsigma ;\epsilon )=\mathfrak{D}(\underset{(n-l)\text{-times}}{\underbrace{\varsigma ,\dots ,\varsigma }},\underset{l\text{-times}}{\underbrace{\epsilon ,\dots ,\epsilon }}).$$

Ashraf et al. [6] introduced the notion of symmetric generalized n-derivations in a ring, building upon the concept of generalized derivation. Let $n\ge 1$ be a fixed positive integer. A symmetric n-linear map $\mathcal{G}:{\mathcal{A}}^n\to \mathcal{A}$ is known to be symmetric linear generalized n-derivation if there exists a symmetric linear n-derivation $\mathfrak{D}:{\mathcal{A}}^n\to \mathcal{A}$ such that $\mathcal{G}\left({\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_i{\varsigma }_i^{\prime },\dots ,{\varsigma }_n\right)=\mathcal{G}\left({\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_i,\dots ,{\varsigma }_n\right){\varsigma }_i^{\prime }+{\varsigma }_i\mathfrak{D}({\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_i^{\prime },\dots ,{\varsigma }_n)$ holds ∀${\varsigma }_i,{\varsigma }_i^{\prime }\in \mathcal{A}″$.

Example 1. 

Let

$$\mathcal{A}=\left\{\left[\begin{array}{cc}a& a\\ 0& 0\end{array}\right]|a\in \mathbb{C}\right\},$$

where $\mathbb{C}$ is a complex field. Next, define an involution ∗ to be the identity map. It is clear that $\mathcal{A}$ is a $C^{\ast }$-algebra under norm defined by $\left|\right|A\left|\right|=\left|a\right|$ for all $A\in \mathcal{A}$. Denote $A_i=\left[\begin{array}{cc}{a}_i& a_i\\ 0& 0\end{array}\right]\in \mathcal{A}$, $a_i\in \mathbb{C}$, $1\le i\le n$, and let us define $\mathcal{G}=\mathfrak{D}:{\mathcal{A}}^n\to \mathcal{A}$ by $\mathfrak{D}(A_1,A_2,\dots ,A_n)=\left[\begin{array}{cc}0& a_1{a}_2\cdots a_n\\ 0& 0\end{array}\right]$ with trace $\mathcal{g}=\mathcal{d}:\mathcal{A}\to \mathcal{A}$ define by $\mathcal{g}\left(\left[\begin{array}{cc}a& a\\ 0& 0\end{array}\right]\right)=\left[\begin{array}{cc}0& a^n\\ 0& 0\end{array}\right]$. Then it is easy to see that $\mathcal{G}$ is a symmetric linear generalized n-derivation on $\mathcal{A}$.

There has been notable scholarly focus on the structure of linear derivations and linear bi-derivations within the context of $C^{\ast }$-algebras. Various authors have provided diverse expositions of derivations on $C^{\ast }$-algebras, showcasing a spectrum of perspectives and methodologies. For instance, Kadison’s work in 1966 [7] demonstrated that every linear derivation acting on a $C^{\ast }$-algebra annihilates its center. In 1989, Mathieu [8] built upon Posner’s first theorem [9] regarding $C^{\ast }$-algebras, extending its implications. Basically, he proved that “if the product of two linear derivations d and $d^{\prime }$ on a $C^{\ast }$-algebra is a linear derivation then $d{d}^{\prime }=0$”. Very recently, Ekrami and Mirzavaziri [10] showed that “if $\mathcal{A}$ is a $C^{\ast }$-algebra admitting two linear derivations d and $d^{\prime }$ on $\mathcal{A}$, then there exists a linear derivation D on $\mathcal{A}$ such that $d{d}^{\prime }+d^{\prime }d=D^2$ if and only if d and $d^{\prime }$ are linearly dependent”.

In [11], Ali and Khan proved that if $\mathcal{A}$ is a $C^{\ast }$-algebra admitting a symmetric bilinear generalized ∗-biderivation $\mathcal{H}:\mathcal{A}\times \mathcal{A}\to \mathcal{A}$ with an associated symmetric bilinear ∗-biderivation $\mathcal{B}:\mathcal{A}\times \mathcal{A}\to \mathcal{A}$, then $\mathcal{H}$ maps $\mathcal{A}\times \mathcal{A}$ into $Z(\mathcal{A})$. In [12], Rehman and Ansari provided a characterization of the trace of symmetric bi-derivations, and they proved more general results by examining different conditions on a subset of the ring R, specifically the Lie ideal of R. Basically, they proved that “let R be a prime ring with $charR\ne 2$ and U be a square closed Lie ideal of R. Suppose that $B:R\times R\to R$ is a symmetric bi-derivation and f, the trace of B. If $[f(x),x]=0$ ∀$x\in U$, then either $U\subseteq Z(R)$ or $f=0$” (see also [13,14,15,16,17,18,19] for recent results).

The motivation behind this research stems from the seminal works of Ali and Khan [11], as well as Rehman and Ansari [12], who explored the intricate connections between bilinear biderivations and algebraic structures within $C^{\ast }$-algebras and prime rings, respectively. In this study, we extend the above mentioned inquiry to the realm of linear generalized n-derivations in $C^{\ast }$-algebras. Focusing specifically on Lie ideals within these algebras, we aim to uncover broader outcomes and novel insights into the intricate relationships between linear generalized n-derivations and algebraic structure of $C^{\ast }$-algebras. By scrutinizing the behavior of linear generalized n-derivations within Lie ideals, our research seeks to elucidate their role in the algebraic landscape, contributing to a deeper understanding of the underlying principles governing linear generalized n-derivations in $C^{\ast }$-algebras. Precisely, we prove that if $\mathcal{A}$ is a $C^{\ast }$-algebra, U is a square closed Lie ideal of $\mathcal{A}$ admitting a nonzero symmetric linear generalized n-derivation $\mathcal{G}:{\mathcal{A}}^n\to \mathcal{A}$ with trace $\mathcal{g}:\mathcal{A}\to \mathcal{A}$ associated with symmetric linear n-derivation $\mathfrak{D}:{\mathcal{A}}^n\to \mathcal{A}$ with trace $\mathcal{d}:\mathcal{A}\to \mathcal{A}$ satisfying the condition $(g(\varsigma \epsilon )-g(\epsilon \varsigma ))\pm [\varsigma ,\epsilon ]\in Z(\mathcal{A})$∀$\varsigma ,\epsilon \in U$, then $U\subseteq Z(\mathcal{A})$.

2. The Results

To initiate the substantiation of our primary theorems, we first articulate a result that we frequently invoke in the demonstration of our principal outcomes.

Lemma 1 

([20], Corollary 2.1). “Let R be a 2-torsion free semiprime ring, U a Lie ideal of R such that $U⊈Z(R)$ and $a,b\in U$.

1. 

If $aUa=\left\{0\right\}$, then $a=0$.

2. 

If $aU=\left\{0\right\}$ ($Ua=\left\{0\right\}$), then $a=0$.

3. 

If U is a square closed Lie ideal and $aUb=\left\{0\right\}$, then $ab=0$ and $ba=0$.

Lemma 2 

([21], Lemma 1). Let R be a semiprime, 2 torsion-free ring and let U be a Lie ideal of R. Suppose that $[U,U]\subseteq Z(R)$, then $U\subseteq Z(R)$.

Lemma 3 

([22]). Let n be a fixed positive integer and R a $n!$-torsion free ring. Suppose that ${\epsilon }_1,{\epsilon }_2,\dots ,{\epsilon }_n\in R$ satisfy $\lambda {\epsilon }_1+{\lambda }^2{\epsilon }_2+\cdots +{\lambda }^n{\epsilon }_n=0$ for $\lambda =1,2,\dots ,n$. Then ${\epsilon }_i=0$ for $i=1,2,\dots ,n$”.

Daif and Bell [23] proved that if a semiprime ring admits a derivation d such that either $\varsigma \epsilon -d(\varsigma \epsilon )=\epsilon \varsigma -d(\epsilon \varsigma )$ or $\varsigma \epsilon +d(\varsigma \epsilon )=\epsilon \varsigma +d(\epsilon \varsigma )$ holds ∀$\varsigma ,\epsilon \in R$, then R is commutative. In this section, apart from proving other results, we expand the previous result by demonstrating the following theorem for the traces of generalized linear n-derivation on well behaved subsets of $\mathcal{A}$.

Theorem 1. 

For any fixed integer $n\ge 2$, let $\mathcal{A}$ be a $C^{\ast }$-algebra, U be a square closed Lie ideal of $\mathcal{A}$. If $\mathcal{A}$ admits a nonzero symmetric linear generalized n-derivation $\mathcal{G}:{\mathcal{A}}^n\to \mathcal{A}$ with trace $\mathcal{g}:\mathcal{A}\to \mathcal{A}$ associated with symmetric linear n-derivation $\mathfrak{D}:{\mathcal{A}}^n\to \mathcal{A}$ with trace $\mathcal{d}:\mathcal{A}\to \mathcal{A}$ satisfying the condition of $(g(\varsigma \epsilon )-g(\epsilon \varsigma ))\pm [\varsigma ,\epsilon ]\in Z(\mathcal{A})$∀$\varsigma ,\epsilon \in U$, then $U\subseteq Z(\mathcal{A})$.

Proof. 

It is given that

$$(g(\varsigma \epsilon )-g(\epsilon \varsigma ))\pm [\varsigma ,\epsilon ]\in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U.$$

Replacing $\epsilon$ by $\varsigma +m\epsilon$, where $1\le m\le n-1$ in the given condition, we obtain

$$\mathcal{g}(\varsigma (\varsigma +m\epsilon ))-\mathcal{g}((\varsigma +m\epsilon )\varsigma )\pm [\varsigma ,\varsigma +m\epsilon ]\in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U$$

which on solving, we have

$$\begin{array}{c}\mathcal{g}(\varsigma m\epsilon )-\mathcal{g}(m\epsilon \varsigma )+{\displaystyle \sum _{l=1}^{n-1}}{}^n{C}_l\mathcal{G}(\underset{(n-l)-\mathrm{times}}{\underbrace{{\varsigma }^2,\dots ,{\varsigma }^2}},\underset{l-\mathrm{times}}{\underbrace{\varsigma m\epsilon ,\dots ,\varsigma m\epsilon }})-{\displaystyle \sum _{l=1}^{n-1}}{}^n{C}_l\mathcal{G}(\underset{(n-l)-\mathrm{times}}{\underbrace{{\varsigma }^2,\dots ,{\varsigma }^2}},\underset{l-\mathrm{times}}{\underbrace{m\epsilon \varsigma ,\dots ,m\epsilon \varsigma }})\hfill \\ \hfill \pm [\varsigma ,m\epsilon ]\in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U.\end{array}$$

(1)

By using hypothesis, we obtain

$${\displaystyle \sum _{l=1}^{n-1}}{}^n{C}_l\mathcal{G}(\underset{(n-l)-\mathrm{times}}{\underbrace{{\varsigma }^2,\dots ,{\varsigma }^2}},\underset{l-\mathrm{times}}{\underbrace{\varsigma m\epsilon ,\dots ,\varsigma m\epsilon }})-{\displaystyle \sum _{l=1}^{n-1}}{}^n{C}_l\mathcal{G}(\underset{(n-l)-\mathrm{times}}{\underbrace{{\varsigma }^2,\dots ,{\varsigma }^2}},\underset{l-\mathrm{times}}{\underbrace{m\epsilon \varsigma ,\dots ,m\epsilon \varsigma }})\pm [\varsigma ,m\epsilon ]\in Z(\mathcal{A})$$

∀$\varsigma ,\epsilon \in U$. Making use of Lemma 3, we see that

$$\mathcal{G}({\varsigma }^2,\dots ,{\varsigma }^2,\varsigma \epsilon )-\mathcal{G}({\varsigma }^2,\dots ,{\varsigma }^2,\epsilon \varsigma )\in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U.$$

(2)

For $1\le m\le n$, (1) can also be written as

$$\begin{array}{c}{m}^n\mathcal{g}(\varsigma \epsilon )-m^n\mathcal{g}(\epsilon \varsigma )+{\displaystyle \sum _{l=1}^{n-1}}{}^n{C}_l\mathcal{G}(\underset{(n-l)-\mathrm{times}}{\underbrace{{\varsigma }^2,\dots ,{\varsigma }^2}},\underset{l-\mathrm{times}}{\underbrace{\varsigma m\epsilon ,\dots ,\varsigma m\epsilon }})-{\displaystyle \sum _{l=1}^{n-1}}{}^n{C}_l\mathcal{G}(\underset{(n-l)-\mathrm{times}}{\underbrace{{\varsigma }^2,\dots ,{\varsigma }^2}},\underset{l-\mathrm{times}}{\underbrace{m\epsilon \varsigma ,\dots ,m\epsilon \varsigma }})\hfill \\ \hfill \pm [\varsigma ,m\epsilon ]\in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U.\end{array}$$

Again making use of Lemma 3, we have

$$n\{\mathcal{G}({\varsigma }^2,\dots ,{\varsigma }^2,\varsigma \epsilon )-\mathcal{G}({\varsigma }^2,\dots ,{\varsigma }^2,\epsilon \varsigma )\}\pm [\varsigma ,\epsilon ]\in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U.$$

(3)

From (2) and (3), we obtain $[\varsigma ,\epsilon ]\in Z(\mathcal{A})$∀$\varsigma ,\epsilon \in U$. As every $C^{\ast }$-algebra is a semiprime ring, using Lemma 2, we obtain $U\subseteq Z(\mathcal{A})$. □

Theorem 2. 

For any fixed integer $n\ge 2$, let $\mathcal{A}$ be a $C^{\ast }$-algebra and U be a square closed Lie ideal of $\mathcal{A}$. If $\mathcal{A}$ admits a nonzero symmetric linear generalized n-derivation $\mathcal{G}:{\mathcal{A}}^n\to \mathcal{A}$ with trace $\mathcal{g}:\mathcal{A}\to \mathcal{A}$ associated with symmetric linear n-derivation $\mathfrak{D}:{\mathcal{A}}^n\to \mathcal{A}$ with trace $\mathcal{d}:\mathcal{A}\to \mathcal{A}$ satisfying the condition of $(g(\varsigma )\pm g(\epsilon ))\pm \varsigma \circ \epsilon \in Z(\mathcal{A})$∀$\varsigma ,\epsilon \in U$, then $U\subseteq Z(\mathcal{A})$.

Proof. 

Suppose on the contrary that $U⊈Z(\mathcal{A})$. We have given that

$$(g(\varsigma )-g(\epsilon ))\pm \varsigma \circ \epsilon \in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U.$$

Replacing $\epsilon$ by $\varsigma +m\epsilon$, where $z\in U$ and $1\le m\le n-1$ in the given condition, we obtain

$$\mathcal{g}(\varsigma )\pm \mathcal{g}(\varsigma +m\epsilon )\pm (\varsigma \circ \varsigma +m\epsilon )\in Z(\mathcal{A})\forall \varsigma ,\epsilon ,z\in U$$

which on solving, we have

$$\begin{array}{c}\hfill \mathcal{g}(\varsigma )\pm \mathcal{g}(\varsigma )\mathcal{g}(m\epsilon )\pm {\displaystyle \sum _{l=1}^{n-1}}{}^n{C}_l\mathcal{G}(\underset{(n-l)-\mathrm{times}}{\underbrace{\varsigma ,\dots ,\varsigma }},\underset{l-\mathrm{times}}{\underbrace{m\epsilon ,\dots ,m\epsilon }})\pm \varsigma \circ \varsigma \pm \varsigma \circ m\epsilon \in Z(\mathcal{A})\forall \varsigma ,\epsilon ,z\in U.\end{array}$$

(4)

Using the given condition, we obtain

$$\mathcal{g}(\varsigma )\pm {\varsigma }^2\pm {\displaystyle \sum _{l=1}^{n-1}}{}^n{C}_l\mathcal{G}(\underset{(n-l)-\mathrm{times}}{\underbrace{\varsigma ,\dots ,\varsigma }},\underset{l-\mathrm{times}}{\underbrace{m\epsilon ,\dots ,m\epsilon }})\in Z(\mathcal{A})\forall \varsigma ,\epsilon ,z\in U.$$

Multiply the above equation by m which implies that

$$m{A}_1(\varsigma ,\epsilon )+m^2{A}_2(\varsigma ,\epsilon )+\cdots +m^{n-1}{A}_{n-1}(\varsigma ,\epsilon )\in Z(\mathcal{A})$$

∀$\varsigma ,\epsilon ,z\in U$ where $A_l(\varsigma ,\epsilon )$ represents the term in which z appears l-times.

Making use of Lemma 3, we see that

$$\mathcal{G}(\varsigma ,\dots ,\varsigma ,\epsilon )\in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U.$$

Replace $\epsilon$ by $\varsigma$, we obtain

$$\mathcal{g}(\varsigma )\in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U.$$

From hypothesis, we have $\varsigma \circ \epsilon \in Z(\mathcal{A})$∀$\varsigma ,\epsilon \in U$. Again replace $\varsigma$ by $\epsilon \varsigma$, we have $\epsilon (\varsigma \circ \epsilon )\in Z(\mathcal{A})$ which imply $[\epsilon (\varsigma \circ \epsilon ),z]\in Z(\mathcal{A})$. On solving, we obtain $[\epsilon ,z](\varsigma \circ \epsilon )=0$∀$\varsigma ,\epsilon ,z\in U$. Again replace $\varsigma$ by $\varsigma z$, we have $[\epsilon ,z]\varsigma [z,\epsilon ]=0$∀$\varsigma ,\epsilon ,z\in U$. By Lemma 1, we have $[z,\epsilon ]=0$∀$\epsilon ,z\in U$. Again using Lemma 2, we obtain $U\subseteq Z(\mathcal{A})$, which is a contradiction. □

Theorem 3. 

Let $\mathcal{A}$ be a $C^{\ast }$-algebra and U be a square closed Lie ideal of $\mathcal{A}$. If $\mathcal{A}$ admits a nonzero symmetric linear generalized n-derivation $\mathcal{G}:{\mathcal{A}}^n\to \mathcal{A}$ with trace $\mathcal{g}:\mathcal{A}\to \mathcal{A}$ associated with symmetric linear n-derivation $\mathfrak{D}:{\mathcal{A}}^n\to \mathcal{A}$ with trace $\mathcal{d}:\mathcal{A}\to \mathcal{A}$ satisfying $\mathcal{g}({\varsigma }^2)\pm {\varsigma }^2=0$∀$\varsigma \in U$, then $U\subseteq Z(\mathcal{A})$.

Proof. 

Suppose on the contrary that $U⊈Z(\mathcal{A})$. We have given that $\mathcal{G}:{\mathcal{A}}^n\to \mathcal{A}$ be symmetric linear generalized n-derivations associated with $\mathfrak{D}:{\mathcal{A}}^n\to \mathcal{A}$ of a $C^{\ast }$-algebra $\mathcal{A}$ such that $\mathcal{g}({\varsigma }^2)\pm {\varsigma }^2=0$∀$\varsigma \in U$. Therefore, $\mathcal{A}$ is semiprime as $\mathcal{A}$ is a $C^{\ast }$-algebra. Now replacing $\varsigma$ by $\varsigma +m\epsilon$, $\epsilon \in U$ for $1\le m\le n-1$ in the given condition, we obtain

$$\mathcal{g}{(\varsigma +m\epsilon )}^2\pm {(\varsigma +m\epsilon )}^2=0\forall \varsigma ,\epsilon \in U.$$

Further solving, we have

$$\begin{array}{c}\mathcal{g}({\varsigma }^2)+\mathcal{g}(m(\varsigma \epsilon +\epsilon \varsigma ))+{\displaystyle \sum _{l=1}^{n-1}}{}^n{C}_l\mathcal{G}(\underset{(n-l)-\mathrm{times}}{\underbrace{{\varsigma }^2,\dots ,{\varsigma }^2}},\underset{l-\mathrm{times}}{\underbrace{m(\varsigma \epsilon +\epsilon \varsigma ),\dots ,m(\varsigma \epsilon +\epsilon \varsigma )}})+\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathcal{g}({(m\epsilon )}^2)+{\displaystyle \sum _{l=1}^{n-1}}{}^n{C}_l\mathcal{G}(\underset{(n-l)-\mathrm{times}}{\underbrace{{\varsigma }^2+m(\varsigma \epsilon +\epsilon \varsigma ),\dots ,{\varsigma }^2+m(\varsigma \epsilon +\epsilon \varsigma )}},\underset{l-\mathrm{times}}{\underbrace{{(m\epsilon )}^2,\dots ,{(m\epsilon )}^2}})\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\pm {\varsigma }^2\pm {(m\epsilon )}^2\pm m(\varsigma \epsilon +\epsilon \varsigma )=0\forall \varsigma ,\epsilon \in U.\end{array}$$

In accordance of the given condition and Lemma 3, we obtain

$$n\mathcal{G}({\varsigma }^2,\dots ,{\varsigma }^2,\varsigma \epsilon +\epsilon \varsigma )\pm (\varsigma \epsilon +\epsilon \varsigma )=0\forall \varsigma ,\epsilon \in U.$$

Replacing $\epsilon$ by $\varsigma$, we find that

$$2n\mathcal{g}({\varsigma }^2)\pm 2{\varsigma }^2=0,$$

or

$${\varsigma }^2=0.$$

This implies that $\varsigma \epsilon +\epsilon \varsigma =0$∀$\varsigma ,\epsilon \in U$. Replacing $\epsilon$ by $\epsilon z$, where $z\in U$, we obtain $[\varsigma ,\epsilon ]z=0$. Again replacing z by $z[\varsigma ,\epsilon ]$, we obtain $[\varsigma ,\epsilon ]z[\varsigma ,\epsilon ]=0$∀$\varsigma ,\epsilon ,z\in U$. Using the Lemma 1, we obtain $[\varsigma ,\epsilon ]=0$∀$\varsigma ,\epsilon \in U$. By Lemma 2, we obtain $U\subseteq Z(\mathcal{A})$, a contradiction. □

Corollary 1. 

For any fixed integer $n\ge 2$, let $\mathcal{A}$ be a $C^{\ast }$-algebra and U be a square closed Lie ideal of $\mathcal{A}$. If $\mathcal{A}$ admits a nonzero symmetric linear generalized n-derivation $\mathcal{G}:{\mathcal{A}}^n\to \mathcal{A}$ with trace $\mathcal{g}:\mathcal{A}\to \mathcal{A}$ associated with symmetric linear n-derivation $\mathfrak{D}:{\mathcal{A}}^n\to \mathcal{A}$ with trace $\mathcal{d}:\mathcal{A}\to \mathcal{A}$ satisfying $\mathcal{g}(\varsigma \circ \epsilon )\pm \varsigma \circ \epsilon =0$∀$\varsigma ,\epsilon \in U$, then $U\subseteq Z(\mathcal{A})$.

Theorem 4. 

For any fixed integer $n\ge 2$, let $\mathcal{A}$ be a $C^{\ast }$-algebra and U be a square closed Lie ideal of $\mathcal{A}$. Let $\mathcal{A}$ admit a nonzero symmetric linear generalized n-derivation $\mathcal{G}:{\mathcal{A}}^n\to \mathcal{A}$ with trace $\mathcal{g}:\mathcal{A}\to \mathcal{A}$ associated with symmetric linear n-derivation $\mathfrak{D}:{\mathcal{A}}^n\to \mathcal{A}$ with trace $\mathcal{d}:\mathcal{A}\to \mathcal{A}$ satisfying one of the following conditions:

(i) 

$[\mathcal{g}(\varsigma ),\mathcal{g}(\epsilon )]-[\varsigma ,\epsilon ]\in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U$

(ii) 

$[\mathcal{g}(\varsigma ),\mathcal{g}(\epsilon )]-[\epsilon ,\varsigma ]\in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U.$

Then, $U\subseteq Z(\mathcal{A})$.

Proof. 

$(i)$ Given that

$$[\mathcal{g}(\varsigma ),\mathcal{g}(\epsilon )]-[\varsigma ,\epsilon ]\in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U.$$

(5)

Consider a positive integer m; $1\le m\le n-1$. Replacing $\epsilon$ by $\epsilon +mz$, where $z\in U$ in (5), we obtain

$$[\mathcal{g}(\varsigma ),\mathcal{g}(\epsilon +mz)]-[\varsigma ,\epsilon +mz]\in Z(\mathcal{A})\forall \varsigma ,\epsilon ,z\in U.$$

On further solving, we obtain

$$\begin{array}{c}[\mathcal{g}(\varsigma ),\mathcal{g}(\epsilon )]+[\mathcal{g}(\varsigma ),\mathcal{g}(mz)]+[\mathcal{g}(\varsigma ),{\displaystyle \sum _{l=1}^{n-1}}{}^n{C}_l\mathcal{G}(\underset{(n-l)-\mathrm{times}}{\underbrace{\epsilon ,\dots ,\epsilon }},\underset{l-\mathrm{times}}{\underbrace{mz,\dots ,mz}})]-\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}[\varsigma ,\epsilon ]-[\varsigma ,mz]\in Z(\mathcal{A})\forall \varsigma ,\epsilon ,z\in U.\end{array}$$

On taking account of hypothesis, we see that

$$m{A}_1(\varsigma ,\epsilon ,z)+m^2{A}_2(\varsigma ,\epsilon ,z)+\cdots +m^{n-1}{A}_{n-1}(\varsigma ,\epsilon ,z)\in Z(\mathcal{A})$$

where $A_l(\varsigma ,\epsilon ,z)$ represents the term in which z appears l-times.

Using Lemma 3, we have

$$[\mathcal{g}(\varsigma ),\mathcal{G}(\epsilon ,\dots ,\epsilon ,z)]\in Z(\mathcal{A})\forall \varsigma ,\epsilon ,z\in U.$$

In particular, for $z=\epsilon$, we obtain

$$[\mathcal{g}(\varsigma ),\mathcal{g}(\epsilon )]\in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U.$$

Now using the given condition, we find that

$$[\varsigma ,\epsilon ]\in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U.$$

From Lemma 2, $U\subseteq Z(\mathcal{A})$.

$(ii)$ Follows from the first implication with a slight modification. □

Corollary 2. 

For any fixed integer $n\ge 2$, let $\mathcal{A}$ be a $C^{\ast }$-algebra and U be a square closed Lie ideal of $\mathcal{A}$. Let $\mathcal{A}$ admit a nonzero symmetric linear generalized n-derivation $\mathcal{G}:{\mathcal{A}}^n\to \mathcal{A}$ with trace $\mathcal{g}:\mathcal{A}\to \mathcal{A}$ associated with symmetric linear n-derivation $\mathfrak{D}:{\mathcal{A}}^n\to \mathcal{A}$ with trace $\mathcal{d}:\mathcal{A}\to \mathcal{A}$ satisfying one of the following conditions:

(i) 

$\mathcal{g}(\varsigma )\mathcal{g}(\epsilon )\pm \varsigma \epsilon \in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U$

(ii) 

$\mathcal{g}(\varsigma )\mathcal{g}(\epsilon )\pm \epsilon \varsigma \in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U.$

Then, $U\subseteq Z(\mathcal{A})$.

Corollary 3. 

For any fixed integer $n\ge 2$, let $\mathcal{A}$ be a $C^{\ast }$-algebra and U be a square closed Lie ideal of $\mathcal{A}$. Let $\mathcal{A}$ admit a nonzero symmetric linear generalized n-derivation $\mathcal{G}:{\mathcal{A}}^n\to \mathcal{A}$ with trace $\mathcal{g}:\mathcal{A}\to \mathcal{A}$ associated with symmetric linear n-derivation $\mathfrak{D}:{\mathcal{A}}^n\to \mathcal{A}$ with trace $\mathcal{d}:\mathcal{A}\to \mathcal{A}$ satisfying one of the following conditions:

(i) 

$[\mathcal{g}(\varsigma ),\mathcal{g}(\epsilon )]=[\varsigma ,\epsilon ]\forall \varsigma ,\epsilon \in U$

(ii) 

$[\mathcal{g}(\varsigma ),\mathcal{g}(\epsilon )]=[\epsilon ,\varsigma ]\forall \varsigma ,\epsilon \in U.$

Then, $U\subseteq Z(\mathcal{A})$.

Theorem 5. 

For any fixed integer $n\ge 2$, let $\mathcal{A}$ be a $C^{\ast }$-algebra and U be a square closed Lie ideal of $\mathcal{A}$. If $\mathcal{A}$ admits a nonzero symmetric linear generalized n-derivation $\mathcal{G}:{\mathcal{A}}^n\to \mathcal{A}$ with trace $\mathcal{g}:\mathcal{A}\to \mathcal{A}$ associated with symmetric linear n-derivation $\mathfrak{D}:{\mathcal{A}}^n\to \mathcal{A}$ with trace $\mathcal{d}:\mathcal{A}\to \mathcal{A}$ satisfying the condition $\mathcal{g}(\varsigma \circ \epsilon )\pm [\varsigma ,\epsilon ]\in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U$, then $U\subseteq Z(\mathcal{A})$.

Proof. 

Replacing $\epsilon$ by $\epsilon +mz$ for $1\le m\le n-1$, $z\in U$ in the given condition, we obtain

$$\mathcal{g}(\varsigma \circ (\epsilon +mz))\pm [\varsigma ,\epsilon +mz]\in Z(\mathcal{A})\forall \varsigma ,\epsilon ,z\in U.$$

On further solving and using the specified condition, we obtain

$${\displaystyle \sum _{l=1}^{n-1}}{}^n{C}_l\mathcal{G}(\underset{(n-l)-\mathrm{times}}{\underbrace{\varsigma \circ \epsilon ,\dots ,\varsigma \circ \epsilon }},\underset{l-\mathrm{times}}{\underbrace{\varsigma \circ mz,\dots ,\varsigma \circ mz}})\in Z(\mathcal{A})\forall \varsigma ,\epsilon ,z\in U$$

which implies that

$$m{A}_1(\varsigma ,\epsilon ,z)+m^2{A}_2(\varsigma ,\epsilon ,z)+\cdots +m^{n-1}{A}_{n-1}(\varsigma ,\epsilon ,z)\in Z(\mathcal{A})$$

∀$\varsigma ,\epsilon ,z\in U$ where $A_l(\varsigma ,\epsilon ,z)$ represents the term in which z appears l-times. Using Lemma 3, we obtain

$$\mathcal{G}(\varsigma \circ \epsilon ,\dots ,\varsigma \circ \epsilon ,\varsigma \circ z)\in Z(\mathcal{A})\forall \varsigma ,\epsilon ,z\in U.$$

(6)

For $z=\epsilon$, we obtain $\mathcal{g}(\varsigma \circ \epsilon )\in Z(\mathcal{A})$ then our hypothesis reduces to $[\varsigma ,\epsilon ]\in Z(\mathcal{A})$. Using the Lemma 2, we obtain $U\subseteq Z(\mathcal{A})$. □

Corollary 4. 

For any fixed integer $n\ge 2$, let $\mathcal{A}$ be a $C^{\ast }$-algebra and U be a square closed Lie ideal of $\mathcal{A}$. If $\mathcal{A}$ admits a nonzero symmetric linear generalized n-derivation $\mathcal{G}:{\mathcal{A}}^n\to \mathcal{A}$ with trace $\mathcal{g}:\mathcal{A}\to \mathcal{A}$ associated with symmetric linear n-derivation $\mathfrak{D}:{\mathcal{A}}^n\to \mathcal{A}$ with trace $\mathcal{d}:\mathcal{A}\to \mathcal{A}$ satisfying the condition $\mathcal{d}(\varsigma \circ \epsilon )\pm [\varsigma ,\epsilon ]\in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U$, then $U\subseteq Z(\mathcal{A})$.

Theorem 6. 

For any fixed integer $n\ge 2$, let $\mathcal{A}$ be a $C^{\ast }$-algebra and U be a square closed Lie ideal of $\mathcal{A}$. Let $\mathcal{A}$ admit a nonzero symmetric linear generalized n-derivation $\mathcal{G}:{\mathcal{A}}^n\to \mathcal{A}$ with trace $\mathcal{g}:\mathcal{A}\to \mathcal{A}$ associated with symmetric linear n-derivation $\mathfrak{D}:{\mathcal{A}}^n\to \mathcal{A}$ with trace $\mathcal{d}:\mathcal{A}\to \mathcal{A}$ satisfying one of the following conditions:

(i) 

$\mathcal{g}([\varsigma ,\epsilon ])\pm \mathcal{g}(\varsigma )\pm [\varsigma ,\epsilon ]\in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U$

(ii) 

$\mathcal{g}([\varsigma ,\epsilon ])\pm \mathcal{g}(\epsilon )\pm [\varsigma ,\epsilon ]\in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U.$

Then, $U\subseteq Z(\mathcal{A})$.

Proof. 

$(i)$ Given that

$$\mathcal{g}([\varsigma ,\epsilon ])\pm \mathcal{g}(\varsigma )\pm [\varsigma ,\epsilon ]\in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U.$$

Replacing $\varsigma$ by $\varsigma +mz$, where $z\in U$ and $1\le m\le n-1$ in the given condition, we obtain

$$\mathcal{g}([\varsigma +mz,\epsilon ])\pm \mathcal{g}(\varsigma +mz)\pm [\varsigma +mz,\epsilon ]\in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U$$

which on solving and using hypothesis, we obtain

$$\begin{array}{c}{\displaystyle \sum _{l=1}^{n-1}}{}^n{C}_l\mathcal{G}(\underset{(n-l)-\mathrm{times}}{\underbrace{[\varsigma ,\epsilon ],\dots ,[\varsigma ,\epsilon ]}},\underset{l-\mathrm{times}}{\underbrace{[mz,\epsilon ],\dots ,[mz,\epsilon ]}})\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\pm {\displaystyle \sum _{l=1}^{n-1}}{}^n{C}_l\mathcal{G}(\underset{(n-l)-\mathrm{times}}{\underbrace{\varsigma ,\dots ,\varsigma }},\underset{l-\mathrm{times}}{\underbrace{mz,\dots ,mz}})\in Z(\mathcal{A})\forall \varsigma ,\epsilon ,z\in U\end{array}$$

which implies that

$$m{A}_1(\varsigma ,\epsilon ,z)+m^2{A}_2(\varsigma ,\epsilon ,z)+\cdots +m^{n-1}{A}_{n-1}(\varsigma ,\epsilon ,z)\in Z(\mathcal{A})$$

∀$\varsigma ,\epsilon ,z\in U$ where $A_l(\varsigma ,\epsilon ,z)$ represents the term in which z appears l-times.

Making use of Lemma 3 and torsion restriction, we see that

$$\mathcal{G}([\varsigma ,\epsilon ],\dots ,[\varsigma ,\epsilon ],[z,\epsilon ])\pm \mathcal{G}(\varsigma ,\dots ,\varsigma ,z)\in Z(\mathcal{A})\forall \varsigma ,\epsilon ,z\in U.$$

Replace z by $\varsigma$ to obtain

$$\mathcal{g}([\varsigma ,\epsilon ])\pm \mathcal{g}(\varsigma )\in Z(\mathcal{A})\forall \varsigma ,\epsilon ,z\in U.$$

Hence, by using the given condition, we find that $[\varsigma ,\epsilon ]\in Z(\mathcal{A})$. On taking account of Lemma 2, we obtain $U\subseteq Z(\mathcal{A})$.

$(ii)$ Given that

$$\mathcal{g}([\varsigma ,\epsilon ])\pm \mathcal{g}(\varsigma )\pm [\varsigma ,\epsilon ]\in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U.$$

Replacing $\epsilon$ by $\epsilon +mz$, where $z\in U$ and $1\le m\le n-1$ in the given condition, we obtain

$$\mathcal{g}([\varsigma ,\epsilon +mz])\pm \mathcal{g}(\epsilon +mz)\pm [\varsigma ,\epsilon +mz]\in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U$$

which on solving and using hypothesis, we obtain

$$\begin{array}{c}{\displaystyle \sum _{l=1}^{n-1}}{}^n{C}_l\mathcal{G}(\underset{(n-l)-\mathrm{times}}{\underbrace{[\varsigma ,\epsilon ],\dots ,[\varsigma ,\epsilon ]}},\underset{l-\mathrm{times}}{\underbrace{[\varsigma ,mz],\dots ,[\varsigma ,mz]}})\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\pm {\displaystyle \sum _{l=1}^{n-1}}{}^n{C}_l\mathcal{G}(\underset{(n-l)-\mathrm{times}}{\underbrace{\epsilon ,\dots ,\epsilon }},\underset{l-\mathrm{times}}{\underbrace{mz,\dots ,mz}})\in Z(\mathcal{A})\forall \varsigma ,\epsilon ,z\in U\end{array}$$

which implies that

$$m{A}_1(\varsigma ,\epsilon ,z)+m^2{A}_2(\varsigma ,\epsilon ,z)+\cdots +m^{n-1}{A}_{n-1}(\varsigma ,\epsilon ,z)\in Z(\mathcal{A})$$

∀$\varsigma ,\epsilon ,z\in U$ where $A_l(\varsigma ,\epsilon ,z)$ represents the term in which z appears l-times.

Making use of Lemma 3 and torsion restriction, we see that

$$\mathcal{G}([\varsigma ,\epsilon ],\dots ,[\varsigma ,\epsilon ],[\varsigma ,z])\pm \mathcal{G}(\epsilon ,\dots ,\epsilon ,z)\in Z(\mathcal{A})\forall \varsigma ,\epsilon ,z\in U.$$

Replace z by $\epsilon$ to obtain

$$\mathcal{g}([\varsigma ,\epsilon ])\pm \mathcal{g}(\epsilon )\in Z(\mathcal{A})\forall \varsigma ,\epsilon ,z\in U.$$

Hence, by using the given condition, we find that $[\varsigma ,\epsilon ]\in Z(\mathcal{A})$. On taking account of Lemma 2, we obtain $U\subseteq Z(\mathcal{A})$.

$(iii)$ Follows from the first implication with a slight modification. □

Corollary 5. 

For any fixed integer $n\ge 2$, let $\mathcal{A}$ be a $C^{\ast }$-algebra and U be a square closed Lie ideal of $\mathcal{A}$. Let $\mathcal{A}$ admit a nonzero symmetric linear n-derivation $\mathfrak{D}:{\mathcal{A}}^n\to \mathcal{A}$ with trace $\mathcal{d}:\mathcal{A}\to \mathcal{A}$ satisfying one of the following conditions:

(i) 

$\mathcal{d}([\varsigma ,\epsilon ])\pm \mathcal{d}(\varsigma )\pm [\varsigma ,\epsilon ]\in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U$

(ii) 

$\mathcal{d}([\varsigma ,\epsilon ])\pm \mathcal{d}(\epsilon )\pm [\varsigma ,\epsilon ]\in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U$

Then, $U\subseteq Z(\mathcal{A})$.

Theorem 7. 

For any fixed integer $n\ge 2$, let $\mathcal{A}$ be a $C^{\ast }$-algebra and U be a square closed Lie ideal of $\mathcal{A}$. Let $\mathcal{A}$ admit a nonzero symmetric linear generalized n-derivation $\mathcal{G}:{\mathcal{A}}^n\to \mathcal{A}$ with trace $\mathcal{g}:\mathcal{A}\to \mathcal{A}$ associated with symmetric linear n-derivation $\mathfrak{D}:{\mathcal{A}}^n\to \mathcal{A}$ with trace $\mathcal{d}:\mathcal{A}\to \mathcal{A}$ satisfying one of the following conditions:

(i) 

$\mathcal{g}(\varsigma )\circ \mathcal{g}(\epsilon )\pm \varsigma \circ \epsilon \in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U$

(ii) 

$\mathcal{g}(\varsigma )\circ \mathcal{g}(\epsilon )\pm [\varsigma ,\epsilon ]\in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U$

Then, $U\subseteq Z(\mathcal{A})$.

Proof. 

$(i)$ Suppose on the contrary that $U⊈Z(\mathcal{A})$. It is given that

$$\mathcal{g}(\varsigma )\circ \mathcal{g}(\epsilon )\pm \varsigma \circ \epsilon \in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U.$$

Replacing $\epsilon$ by $\epsilon +mz$, where $z\in U$ and $1\le m\le n-1$ in the given condition, we obtain

$$\mathcal{g}(\varsigma )\circ \mathcal{g}(\epsilon +mz)\pm \varsigma \circ (\epsilon +mz)\in Z(\mathcal{A})\forall \varsigma ,\epsilon ,z\in U$$

which on solving, we have

$$\begin{array}{c}\mathcal{g}(\varsigma )\circ \mathcal{g}(\epsilon )+\mathcal{g}(\varsigma )\circ \mathcal{g}(mz)+\mathcal{g}(\varsigma )\circ {\displaystyle \sum _{l=1}^{n-1}}{}^n{C}_l\mathcal{G}(\underset{(n-l)-\mathrm{times}}{\underbrace{\epsilon ,\dots ,\epsilon }},\underset{l-\mathrm{times}}{\underbrace{mz,\dots ,mz}})\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\pm \varsigma \circ \epsilon \pm \varsigma \circ mz\in Z(\mathcal{A})\forall \varsigma ,\epsilon ,z\in U.\end{array}$$

By using hypothesis, we obtain

$$\mathcal{g}(\varsigma )\circ {\displaystyle \sum _{l=1}^{n-1}}{}^n{C}_l\mathfrak{D}(\underset{(n-l)-\mathrm{times}}{\underbrace{\epsilon ,\dots ,\epsilon }},\underset{l-\mathrm{times}}{\underbrace{mz,\dots ,mz}})\in Z(\mathcal{A})\forall \varsigma ,\epsilon ,z\in U$$

which implies that

$$m{A}_1(\varsigma ,\epsilon ,z)+m^2{A}_2(\varsigma ,\epsilon ,z)+\cdots +m^{n-1}{A}_{n-1}(\varsigma ,\epsilon ,z)\in Z(\mathcal{A})$$

∀$\varsigma ,\epsilon ,z\in U$ where $A_l(\varsigma ,\epsilon ,z)$ represents the term in which z appears l-times.

Making use of Lemma 3, we see that

$$\mathcal{g}(\varsigma )\circ \mathcal{G}(\epsilon ,\dots ,\epsilon ,z)\in Z(\mathcal{A})\forall \varsigma ,\epsilon ,z\in U.$$

In particular, $z=\epsilon$, we obtain

$$\mathcal{g}(\varsigma )\circ \mathcal{g}(\epsilon )\in Z(\mathcal{A})\forall \varsigma ,\epsilon \in U.$$

Hence, by using the given condition, we find that $\varsigma \circ \epsilon \in Z(\mathcal{A})$∀$\varsigma ,\epsilon \in U$. Replacing $\varsigma$ by $\epsilon \varsigma$, we obtain $\epsilon (\varsigma \circ \epsilon )\in Z(\mathcal{A})$∀$\varsigma ,\epsilon \in U$. We can also write it as

$$[\epsilon (\varsigma \circ \epsilon ),z]\forall \varsigma ,\epsilon ,z\in U$$

which on solving, we obtain $[\epsilon ,z]\varsigma \circ \epsilon =0$∀$\varsigma ,\epsilon ,z\in U$. Again replace $\varsigma$ by $\varsigma z$ and using the same equation, we obtain $[\epsilon ,z]\varsigma [z,\epsilon ]=0$∀$\varsigma ,\epsilon ,z\in U$. Using Lemma 1, we have $[z,\epsilon ]=0$∀$z,\epsilon \in U$. By Lemma 2, we have $U\subseteq Z(\mathcal{A})$ which is a contradiction.

$(ii)$ Proceeding in the same way as in $(i)$, we conclude. □

Corollary 6. 

For any fixed integer $n\ge 2$, let $\mathcal{A}$ be a $C^{\ast }$-algebra and U be a square closed Lie ideal of $\mathcal{A}$. Let $\mathcal{A}$ admit a nonzero symmetric linear n-derivation $\mathfrak{D}:{\mathcal{A}}^n\to \mathcal{A}$ with trace $\mathcal{d}:\mathcal{A}\to \mathcal{A}$ satisfying one of the following conditions:

(i) 

$\mathcal{d}(\varsigma )\circ \mathcal{d}(\epsilon )\pm \varsigma \circ \epsilon =0\forall \varsigma ,\epsilon \in U$

(ii) 

$\mathcal{d}(\varsigma )\circ \mathcal{d}(\epsilon )\pm [\varsigma ,\epsilon ]=0\forall \varsigma ,\epsilon \in U.$

Then, $U\subseteq Z(\mathcal{A})$.

3. Conclusions

In this study, we have explored the structural properties of $C^{\ast }$-algebras through the lens of generalized linear n-derivations. In fact, our investigation delves into the structure of $C^{\ast }$-algebras, focusing particularly on the intricate interplay between symmetric generalized n-derivations $\mathcal{A}$ and Lie ideals of $\mathcal{A}$. By elucidating the functional identity governing the behavior of linear generalized n-derivations, we provided insights into their forms of traces, thus shedding light on their intrinsic properties and behaviors.