Linear Generalized N-derivations on C-algebras*

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Linear Generalized N-derivations on C-algebras*

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Submitted:

10 April 2024

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Abstract

Let $n \geq 2$ be a fixed integer and $\mathcal{A}$ be a $C^*$-algebra. A permuting $ n $-linear map $ \mathcal{G} : \mathcal{A} ^{n} \rightarrow \mathcal{A} $ is known to be symmetric generalized $n$-derivation if there exists a symmetric $n$-derivation $ \mathfrak{D} : \mathcal{A} ^{n} \rightarrow \mathcal{A} $ such that $ \mathcal{G} \left(x_{1}, x_{2}, \ldots, x_{i} x_{i}^{\prime}, \ldots, x_{n}\right)= \mathcal{G} \left(x_{1}, x_{2}, \ldots, x_{i}, \ldots, x_{n}\right) x_{i}^{\prime}+x_{i} \mathfrak{D} (x_{1}, x_{2},\ldots, x_{i}^{\prime}, \ldots, x_{n})$ holds for all $x_{i}, x_{i}^{\prime} \in \mathcal{A} $. In this paper, we investigate the structure of $C^*$-algebras involving generalized linear $n$-derivations. Moreover, we describe the forms of traces of linear $n$-derivations satisfying certain functional identity.

Keywords:

Subject: Computer Science and Mathematics - Algebra and Number Theory

MSC: 47B48; 22D25; 46L55; 16W25

2. Introduction

Throughout the discussion, unless otherwise mentioned,

A

will denote

C^{*}

-algebra with

Z (A)

as its center. However,

A

may or may not have unity. The symbols

[x, y]

and

x \circ y

denote the commutator

x y - y x

and the anti-commutator

x y + y x

, respectively, for any

x, y \in A

. An algebra

A

is said to be prime if

x A y = {0}

implies that either

x = 0

y = 0

, and semiprime if

x A x = {0}

implies that

x = 0

, where

x, y \in A

. An additive subgroup U of R is said to be a Lie ideal of R if

[u, r] \in U

, for all

u \in U

r \in R

. U is called a square-closed Lie ideal of R if U is a Lie ideal and

u^{2} \in U

for all

u \in U

A Banach algebra is a linear associate algebra which, as a vector space, is a Banach space with norm

| | . | |

satisfying the multiplicative inequality;

| | x y | | \leq | | x | | | | y | |

for all x and y in

A

. An involution on an algebra

A

is a linear map

x \mapsto x^{*}

A

into itself such that the following conditions are hold:

(i)

{(x y)}^{*} = y^{*} x^{*}

(i i)

{(x^{*})}^{*} = x

, and

(i i i)

{(x + λ y)}^{*} = x^{*} + \bar{λ} y^{*}

for all

x, y \in A

and

λ \in C

, the field of complex numbers, where

\bar{λ}

is the conjugate of

λ

. An algebra equipped with an involution is called a *-algebra or algebra with involution.

A Banach *-algebra is a Banach algebra

A

together with an isometric involution

| | x^{*} | | = | | x | |

for all

x \in A

. A

C^{*}

-algebra

A

is a Banach *-algebra with the additional norm condition

| | x^{*} x | | = {| | x | |}^{2}

for all

x \in A

A linear operator

D

on a

C^{*}

-algebra

A

is called a derivation if

D (x y) = D (x) y + x D (y)

holds for all

x, y \in A

. Consider the inner derivation

δ_{a}

implemented by an element a in

A

, which is defined as

δ_{a} (x) = x a - a x

for every x in

A

, as a typical example of a nonzero derivation in a noncommutative algebra.

In order to broaden the scope of derivation, Maksa [13] introduced the concept of symmetric bi-derivations. A bi-linear map

D : A \times A \to A

is said to be a bi-derivation if

D (x x^{'}, y) = D (x, y) x^{'} + x D (x^{'}, y)

D (x, y y^{'}) = D (x, y) y^{'} + y D (x, y^{'})

holds for any

x, x^{'}, y, y^{'} \in A

. The foregoing conditions are identical if

D

is also a symmetric map, that is, if

D (x, y) = D (y, x)

for every

x, y \in A

. In this case,

D

is referred to as a symmetric bi-derivation of

A

. Vukman [20] investigated symmetric bi-derivations in prime and semiprime rings. Argac and Yenigul [3] and Muthana [15] obtained the similar type of results on Lie ideals of R.

In this paper we briefly discuss the various extensions of the notion of derivations on

C^{*}

-algebras. The most general and important one among them is the notion of a generalized symmetric linear n-derivations on

C^{*}

-algebras. The concept of derivation and symmetric bi-derivation was generalized by Park [16] as follows: a n-linear map

D : A^{n} \to A

is said to be a symmetric(permuting) linear n-derivation if

D

is permuting and

D (x_{1}, x_{2}, \dots, x_{i} x_{i}^{'}, \dots, x_{n}) = D (x_{1}, x_{2}, \dots, x_{n}) x_{i}^{'} + x_{i} D (x_{1}^{'}, x_{2}, \dots, x_{n})

hold for all

x_{i}, x_{i}^{'} \in A, i = 1, 2, \dots, n

. A map

d : A \to A

defined by

d (x) = D (x, x, \dots, x)

is called the trace of

D

. If

D : A^{n} \to A

is permuting and n-linear, then the trace

d

D

satisfies the relation

d (x + y) = d (x) + d (y) + {\sum^{n}}_{k = 1}^{n - 1} C_{k} h_{k} (x; y)

for all

x, y \in A

, where

n C_{k} = (\binom{n}{k})

and

h_{k} (x; y) = D (\underset{(n - k) - times}{\underset{︸}{x, \dots, x}}, \underset{k - times}{\underset{︸}{y, \dots, y}}) .

Ashraf et al. [4] introduced the notion of symmetric generalized n-derivations in a ring, building upon the concept of generalized derivation. Let

n \geq 1

be a fixed positive integer. A symmetric n-linear map

G : A^{n} \to A

is known to be symmetric linear generalized n-derivation if there exists a symmetric linear n-derivation

D : A^{n} \to A

such that

G (x_{1}, x_{2}, \dots, x_{i} x_{i}^{'}, \dots, x_{n}) = G (x_{1}, x_{2}, \dots, x_{i}, \dots, x_{n}) x_{i}^{'} + x_{i} D (x_{1}, x_{2}, \dots, x_{i}^{'}, \dots, x_{n})

holds for all

x_{i}, x_{i}^{'} \in A

There has been considerable interest in the structure of linear derivation and linear bi-derivation on

C^{*}

-algebras. Derivations on

C^{*}

-algebras were described in various ways by different authors. For example, in 1966, Kadison [11] proved that each linear derivation of a

C^{*}

-algebra annihilates its center. In 1989, Mathieu [14] extended the Posner’s first result [17] on

C^{*}

-algebras. Basically, he proved that if the product of two linear derivations d and

d^{'}

on a

C^{*}

-algebra is a linear derivation then

d d^{'} = 0

. Very recently, Ekrami and Mirzavaziri [7] showed that “if

A

is a

C^{*}

-algebra admitting two linear derivations d and

d^{'}

A

, then there exists a linear derivation D on

A

such that

d d^{'} + d^{'} d = D^{2}

if and only if d and

d^{'}

are linearly dependent".

In [2], Ali and Khan proved that if

A

is a

C^{*}

-algebra admitting a symmetric bilinear generalized *-biderivation

H : A \times A \to A

with an associated symmetric bilinear *-biderivation

B : A \times A \to A

, then

H

maps

A \times A

into

Z (A)

. In [19], Rehman and Ansari characterized the trace of symmetric bi-derivation and obtain more general results by considering various conditions on a subset of the ring R, viz. Lie ideal of R. Basically, they proved that: let R be a prime ring with

c h a r R \neq 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

for all

x \in U

, then either

U \subseteq Z (R)

f = 0

(see also [1,8,12,18] for recent results).

In the prospect of above motivation, we try to prove some results based on linear generalized n-derivations of

C^{*}

-algebras. We aim to achieve broader outcomes by examining various conditions within a specific subset of the

C^{*}

-algebra

A

, viz. Lie ideal of

A

. Precisely, we prove that if

A

is a

C^{*}

-algebra, U is a square closed Lie ideal of

A

admitting a nonzero symmetric linear generalized n-derivation

G : A^{n} \to A

with trace

g : A \to A

associated with symmetric linear n-derivation

D : A^{n} \to A

with trace

d : A \to A

satisfying the condition of

(g (x y) - g (y x)) \pm [x, y] \in Z (R)

for all

x, y \in U

, then

U \subseteq Z (R)

3. The Results

In order to establish the proofs of our main theorems, we first state a result which we use frequently in the proof of our main results.

Lemma 1.

[10, Corrolary 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

If $a U a = {0}$ , then $a = 0$ .
If $a U = {0}$ ( $U a = {0}$ ), then $a = 0$ .
If U is a square closed Lie ideal and $a U b = {0}$ , then $a b = 0$ and $b a = 0$ .

Lemma 2.

[9, 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.

[5] Let n be a fixed positive integer and R a

n!

-torsion free ring. Suppose that

y_{1}, y_{2}, \dots, y_{n} \in R

satisfy

λ y_{1} + λ^{2} y_{2} + \dots + λ^{n} y_{n} = 0

for

λ = 1, 2, \dots, n

. Then

y_{i} = 0

for

i = 1, 2, \dots, n .

Daif and Bell [6] proved that if a semiprime ring admits a derivation d such that either

x y - d (x y) = y x - d (y x)

x y + d (x y) = y x + d (y x)

holds for all

x, y \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 certain subsets of

A

Theorem 1.

For any fixed integer

n \geq 2

, let

A

be a

C^{*}

-algebra, U be a square closed Lie ideal of

A

. If

A

admits a nonzero symmetric linear generalized n-derivation

G : A^{n} \to A

with trace

g : A \to A

associated with symmetric linear n-derivation

D : A^{n} \to A

with trace

d : A \to A

satisfying the condition of

(g (x y) - g (y x)) \pm [x, y] \in Z (A)

for all

x, y \in U

, then

U \subseteq Z (A)

Proof.

It is given that

(g (x y) - g (y x)) \pm [x, y] \in Z (A) f o r a l l x, y \in U .

Replacing y by

x + m y

, where

1 \leq m \leq n - 1

in the given condition, we get

g (x (x + m y)) - g ((x + m y) x) \pm [x, x + m y] \in Z (A) f o r a l l x, y \in U

which on solving, we have

\begin{matrix} g (x m y) - g (m y x) + {\sum^{n}}_{t = 1}^{n - 1} C_{t} G (\underset{(n - t) - times}{\underset{︸}{x^{2}, \dots, x^{2}}}, \underset{t - times}{\underset{︸}{x m y, \dots, x m y}}) - {\sum^{n}}_{t = 1}^{n - 1} C_{t} G (\underset{(n - t) - times}{\underset{︸}{x^{2}, \dots, x^{2}}}, \underset{t - times}{\underset{︸}{m y x, \dots, m y x}}) \\ \pm [x, m y] \in Z (A) f o r a l l x, y \in U . \end{matrix}

(1)

By using hypothesis, we get

{\sum^{n}}_{t = 1}^{n - 1} C_{t} G (\underset{(n - t) - times}{\underset{︸}{x^{2}, \dots, x^{2}}}, \underset{t - times}{\underset{︸}{x m y, \dots, x m y}}) - {\sum^{n}}_{t = 1}^{n - 1} C_{t} G (\underset{(n - t) - times}{\underset{︸}{x^{2}, \dots, x^{2}}}, \underset{t - times}{\underset{︸}{m y x, \dots, m y x}}) \pm [x, m y] \in Z (A)

for all

x, y \in U

. Making use of Lemma 3, we see that

G (x^{2}, \dots, x^{2}, x y) - G (x^{2}, \dots, x^{2}, y x) \in Z (A) f o r a l l x, y \in U .

(2)

For

1 \leq m \leq n

, (1) can also be written as

\begin{matrix} m^{n} g (x y) - m^{n} g (y x) + {\sum^{n}}_{t = 1}^{n - 1} C_{t} G (\underset{(n - t) - times}{\underset{︸}{x^{2}, \dots, x^{2}}}, \underset{t - times}{\underset{︸}{x m y, \dots, x m y}}) - {\sum^{n}}_{t = 1}^{n - 1} C_{t} G (\underset{(n - t) - times}{\underset{︸}{x^{2}, \dots, x^{2}}}, \underset{t - times}{\underset{︸}{m y x, \dots, m y x}}) \\ \pm [x, m y] \in Z (A) f o r a l l x, y \in U . \end{matrix}

Agai making use of Lemma 3, we have

n {G (x^{2}, \dots, x^{2}, x y) - G (x^{2}, \dots, x^{2}, y x)} \pm [x, y] \in Z (A) f o r a l l x, y \in U .

(3)

From (2) and (3), we get

[x, y] \in Z (A)

for all

x, y \in U

. As every

C^{*}

-algebra is a semiprime ring, using Lemma 2, we get

U \subseteq Z (A)

. □

Theorem 2.

For any fixed integer

n \geq 2

, let

A

be a

C^{*}

-algebra, U be a square closed Lie ideal of

A

. If

A

admits a nonzero symmetric linear generalized n-derivation

G : A^{n} \to A

with trace

g : A \to A

associated with symmetric linear n-derivation

D : A^{n} \to A

with trace

d : A \to A

satisfying the condition of

(g (x) \pm g (y)) \pm x \circ y \in Z (A)

for all

x, y \in U

, then

U \subseteq Z (A)

Proof.

Suppose on the contrary that

U ⊈ Z (A)

. We have given that

(g (x) - g (y)) \pm x \circ y \in Z (A) f o r a l l x, y \in U .

Replacing y by

x + m y

, where

z \in U

and

1 \leq m \leq n - 1

in the given condition, we get

g (x) \pm g (x + m y) \pm (x \circ x + m y) \in Z (A) f o r a l l x, y, z \in U

which on solving, we have

\begin{matrix} g (x) \pm g (x) g (m y) \pm {\sum^{n}}_{t = 1}^{n - 1} C_{t} G (\underset{(n - t) - times}{\underset{︸}{x, \dots, x}}, \underset{t - times}{\underset{︸}{m y, \dots, m y}}) \pm x \circ x \pm x \circ m y \in Z (A) f o r a l l x, y, z \in U . \end{matrix}

(4)

Using the given condition, we get

g (x) \pm x^{2} \pm {\sum^{n}}_{t = 1}^{n - 1} C_{t} G (\underset{(n - t) - times}{\underset{︸}{x, \dots, x}}, \underset{t - times}{\underset{︸}{m y, \dots, m y}}) \in Z (A) f o r a l l x, y, z \in U .

Multiply the above equation by m which implies that

m A_{1} (x, y) + m^{2} A_{2} (x, y) + \dots + m^{n - 1} A_{n - 1} (x, y) \in Z (A)

for all

x, y, z \in U

where

A_{t} (x, y)

represents the term in which z appears t-times.

Making use of Lemma 3, we see that

G (x, \dots, x, y) \in Z (A) f o r a l l x, y \in U .

Replace y by x, we get

g (x) \in Z (A) f o r a l l x, y, \in U .

From hypothesis, we have

x \circ y \in Z (A)

for all

x, y \in U

. Again replace x by

y x

, we have

y (x \circ y) \in Z (A)

which imply

[y (x \circ y), z] \in Z (A)

. On solving, we get

[y, z] (x \circ y) = 0

for all

x, y, x \in U

. Again replace x by

x z

, we have

[y, z] x [z, y] = 0

for all

x, y, z \in U

. By Lemma 1, we have

[z, y] = 0

for all

y, z \in U

. Again using Lemma 2, we get

U \subseteq Z (A)

, which is a contradiction. □

Theorem 3.

Let

A

be a

C^{*}

-algebra, U be a square closed Lie ideal of

A

. Suppose that

A

admits a nonzero symmetric linear generalized n-derivation

G : A^{n} \to A

with trace

g : A \to A

associated with symmetric linear n-derivation

D : A^{n} \to A

with trace

d : A \to A

satisfying

g (x^{2}) \pm x^{2} = 0

for all

x \in U

, then

U \subseteq Z (A)

Proof.

Suppose on the contrary that

U ⊈ Z (A)

. We have given that

G : A^{n} \to A

be symmetric linear generalized n-derivations associated with

D : A^{n} \to A

of a

C^{*}

-algebra

A

such that

g (x^{2}) \pm x^{2} = 0

for all

x \in U

. Therefore,

A

is semiprime as

A

is a

C^{*}

-algebra. Now replacing x by

x + m y

y \in U

for

1 \leq m \leq n - 1

in the given condition, we get

g {(x + m y)}^{2} \pm {(x + m y)}^{2} = 0 f o r a l l x, y \in U .

Further solving, we have

\begin{matrix} g (x^{2}) + g (m (x y + y x)) + {\sum^{n}}_{t = 1}^{n - 1} C_{t} G (\underset{(n - t) - times}{\underset{︸}{x^{2}, \dots, x^{2}}}, \underset{t - times}{\underset{︸}{m (x y + y x), \dots, m (x y + y x)}}) + \\ g ({(m y)}^{2}) + {\sum^{n}}_{t = 1}^{n - 1} C_{t} G (\underset{(n - t) - times}{\underset{︸}{x^{2} + m (x y + y x), \dots, x^{2} + m (x y + y x)}}, \underset{t - times}{\underset{︸}{{(m y)}^{2}, \dots, {(m y)}^{2}}}) \\ \pm x^{2} \pm {(m y)}^{2} \pm m (x y + y x) = 0 f o r a l l x, y \in U . \end{matrix}

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

n G (x^{2}, \dots, x^{2}, x y + y x) \pm (x y + y x) = 0 f o r a l l x, y \in U .

Replacing y by x, we find that

2 n g (x^{2}) \pm 2 x^{2} = 0,

x^{2} = 0 .

This implies that

x y + y x = 0

for all

x, y \in U

. Replacing y by

y z

, where

z \in U

, we get

[x, y] z = 0

. Again replacing z by

z [x, y]

, we get

[x, y] z [x, y] = 0

for all

x, y, z \in U

. Using the Lemma 1, we get

[x, y] = 0

for all

x, y \in U

. By Lemma 2, we get

U \subseteq Z (A)

, a contradiction. □

Corollary 1.

For any fixed integer

n \geq 2

, let

A

be a

C^{*}

-algebra, U be a square closed Lie ideal of

A

. If

A

admits a nonzero symmetric linear generalized n-derivation

G : A^{n} \to A

with trace

g : A \to A

associated with symmetric linear n-derivation

D : A^{n} \to A

with trace

d : A \to A

satisfying

g (x \circ y) \pm x \circ y = 0

for all

x, y \in U

then

U \subseteq Z (A)

Theorem 4.

For any fixed integer

n \geq 2

, let

A

be a

C^{*}

-algebra, U be a square closed Lie ideal of

A

. If

A

admits a nonzero symmetric linear generalized n-derivation

G : A^{n} \to A

with trace

g : A \to A

associated with symmetric linear n-derivation

D : A^{n} \to A

with trace

d : A \to A

satisfying one of the following conditions:

(i): $[g (x), g (y)] - [x, y] \in Z (A) f o r a l l x, y \in U$
(ii): $[g (x), g (y)] - [y, x] \in Z (A) f o r a l l x, y \in U .$

Then

U \subseteq Z (A)

Proof. (i) Given that

[g (x), g (y)] - [x, y] \in Z (A) f o r a l l x, y \in U .

(5)

Consider a positive integer m;

1 \leq m \leq n - 1

. Replacing y by

y + m z

, where

z \in U

in (5), we get

[g (x), g (y + m z)] - [x, y + m z] \in Z (A) f o r a l l x, y, z \in U .

On further solving, we get

\begin{matrix} [g (x), g (y)] + [g (x), g (m z)] + [g (x), {\sum^{n}}_{t = 1}^{n - 1} C_{t} G (\underset{(n - t) - times}{\underset{︸}{y, \dots, y}}, \underset{t - times}{\underset{︸}{m z, \dots, m z}})] - \\ - [x, m z] \in Z (A) f o r a l l x, y, z \in U . \end{matrix}

On taking account of hypothesis, we see that

m A_{1} (x, y, z) + m^{2} A_{2} (x, y, z) + \dots + m^{n - 1} A_{n - 1} (x, y, z) \in Z (A)

where

A_{t} (x, y, z)

represents the term in which z appears t-times.

Using Lemma 3, we have

[g (x), G (y, \dots, y, z)] \in Z (A) f o r a l l x, y, z \in U .

In particular, for

z = y

, we get

[g (x), g (y)] \in Z (A) f o r a l l x, y \in U .

Now using the given condition, we find that

[x, y] \in Z (A) f o r a l l x, y \in U .

From Lemma 2,

U \subseteq Z (A)

(i i)

Follows from the first implication with a slight modification. □

Corollary 2.

For any fixed integer

n \geq 2

, let

A

be a

C^{*}

-algebra, U be a square closed Lie ideal of

A

. If

A

admits a nonzero symmetric linear generalized n-derivation

G : A^{n} \to A

with trace

g : A \to A

associated with symmetric linear n-derivation

D : A^{n} \to A

with trace

d : A \to A

satisfying one of the following conditions:

(i): $g (x) g (y) \pm x y \in Z (A) f o r a l l x, y \in U$
(ii): $g (x) g (y) \pm y x \in Z (A) f o r a l l x, y \in U .$

Then

U \subseteq Z (A)

Corollary 3.

For any fixed integer

n \geq 2

, let

A

be a

C^{*}

-algebra, U be a square closed Lie ideal of

A

. If

A

admits a nonzero symmetric linear generalized n-derivation

G : A^{n} \to A

with trace

g : A \to A

associated with symmetric linear n-derivation

D : A^{n} \to A

with trace

d : A \to A

satisfying one of the following conditions:

(i): $[g (x), g (y)] = [x, y] f o r a l l x, y \in U$
(ii): $[g (x), g (y)] = [y, x] f o r a l l x, y \in U .$

Then

U \subseteq Z (A)

Theorem 5.

For any fixed integer

n \geq 2

, let

A

be a

C^{*}

-algebra, U be a square closed Lie ideal of

A

. If

A

admits a nonzero symmetric linear generalized n-derivation

G : A^{n} \to A

with trace

g : A \to A

associated with symmetric linear n-derivation

D : A^{n} \to A

with trace

d : A \to A

satisfying the condition

g (x \circ y) \pm [x, y] \in Z (A) f o r a l l x, y \in U

, then

U \subseteq Z (A)

Proof.

Replacing y by

y + m z

for

1 \leq m \leq n - 1

z \in U

in the given condition, we get

g (x \circ (y + m z)) \pm [x, y + m z] \in Z (A) f o r a l l x, y, z \in U .

On further solving and using the specified condition, we get

{\sum^{n}}_{t = 1}^{n - 1} C_{t} G (\underset{(n - t) - times}{\underset{︸}{x \circ y, \dots, x \circ y}}, \underset{t - times}{\underset{︸}{x \circ m z, \dots, x \circ m z}}) \in Z (A) f o r a l l x, y, z \in U

which implies that

m A_{1} (x, y, z) + m^{2} A_{2} (x, y, z) + \dots + m^{n - 1} A_{n - 1} (x, y, z) \in Z (A)

for all

x, y, z \in U

where

A_{t} (x, y, z)

represents the term in which z appears t-times. Using Lemma 3, we get

G (x \circ y, \dots, x \circ y, x \circ z) \in Z (A) f o r a l l x, y, z \in U .

(6)

For

z = y

, we get

g (x \circ y) \in Z (A)

then our hypothesis reduces to

[x, y] \in Z (A)

. Using the Lemma 2, we get

U \subseteq Z (A)

. □

Corollary 4.

For any fixed integer

n \geq 2

, let

A

be a

C^{*}

-algebra, U be a square closed Lie ideal of

A

. If

A

admits a nonzero symmetric linear generalized n-derivation

G : A^{n} \to A

with trace

g : A \to A

associated with symmetric linear n-derivation

D : A^{n} \to A

with trace

d : A \to A

satisfying the condition

d (x \circ y) \pm [x, y] \in Z (A) f o r a l l x, y \in U

, then

U \subseteq Z (A)

Theorem 6.

For any fixed integer

n \geq 2

, let

A

be a

C^{*}

-algebra, U be a square closed Lie ideal of

A

. If

A

admits a nonzero symmetric linear generalized n-derivation

G : A^{n} \to A

with trace

g : A \to A

associated with symmetric linear n-derivation

D : A^{n} \to A

with trace

d : A \to A

satisfying one of the following conditions:

(i): $g ([x, y]) \pm g (x) \pm [x, y] \in Z (A) f o r a l l x, y \in U$
(ii): $g ([x, y]) \pm g (y) \pm [x, y] \in Z (A) f o r a l l x, y \in U .$

Then

U \subseteq Z (A)

Proof.

(i)

Given that

g ([x, y]) \pm g (x) \pm [x, y] \in Z (A) f o r a l l x, y \in U .

Replacing x by

x + m z

, where

z \in U

and

1 \leq m \leq n - 1

in the given condition, we get

g ([x + m z, y]) \pm g (x + m z) \pm [x + m z, y] \in Z (A) f o r a l l x, y \in U

which on solving and using hypothesis, we obtain

\begin{matrix} {\sum^{n}}_{t = 1}^{n - 1} C_{t} G (\underset{(n - t) - times}{\underset{︸}{[x, y], \dots, [x, y]}}, \underset{t - times}{\underset{︸}{[m z, y], \dots, [m z, y]}}) \\ \pm {\sum^{n}}_{t = 1}^{n - 1} C_{t} G (\underset{(n - t) - times}{\underset{︸}{x, \dots, x}}, \underset{t - times}{\underset{︸}{m z, \dots, m z}}) \in Z (A) f o r a l l x, y, z \in U \end{matrix}

which implies that

m A_{1} (x, y, z) + m^{2} A_{2} (x, y, z) + \dots + m^{n - 1} A_{n - 1} (x, y, z) \in Z (A)

for all

x, y, z \in U

where

A_{t} (x, y, z)

represents the term in which z appears t-times.

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

G ([x, y], \dots, [x, y], [z, y]) \pm G (x, \dots, x, z) \in Z (A) f o r a l l x, y, z \in U .

Replace z by x to get

g ([x, y]) \pm g (x) \in Z (A) f o r a l l x, y, z \in U .

Hence, by using the given condition, we find that

[x, y] \in Z (A)

. On taking account of Lemma 2, we get

U \subseteq Z (A)

(i i)

Given that

g ([x, y]) \pm g (x) \pm [x, y] \in Z (A) f o r a l l x, y \in U .

Replacing y by

y + m z

, where

z \in U

and

1 \leq m \leq n - 1

in the given condition, we get

g ([x, y + m z]) \pm g (y + m z) \pm [x, y + m z] \in Z (A) f o r a l l x, y \in U

which on solving and using hypothesis, we obtain

\begin{matrix} {\sum^{n}}_{t = 1}^{n - 1} C_{t} G (\underset{(n - t) - times}{\underset{︸}{[x, y], \dots, [x, y]}}, \underset{t - times}{\underset{︸}{[x, m z], \dots, [x, m z]}}) \\ \pm {\sum^{n}}_{t = 1}^{n - 1} C_{t} G (\underset{(n - t) - times}{\underset{︸}{y, \dots, y}}, \underset{t - times}{\underset{︸}{m z, \dots, m z}}) \in Z (A) f o r a l l x, y, z \in U \end{matrix}

which implies that

m A_{1} (x, y, z) + m^{2} A_{2} (x, y, z) + \dots + m^{n - 1} A_{n - 1} (x, y, z) \in Z (A)

for all

x, y, z \in U

where

A_{t} (x, y, z)

represents the term in which z appears t-times.

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

G ([x, y], \dots, [x, y], [x, z]) \pm G (y, \dots, y, z) \in Z (A) f o r a l l x, y, z \in U .

Replace z by y to get

g ([x, y]) \pm g (y) \in Z (A) f o r a l l x, y, z \in U .

Hence, by using the given condition, we find that

[x, y] \in Z (A)

. On taking account of Lemma 2, we get

U \subseteq Z (A)

(i i)

Follows from the first implication with a slight modification. □

Corollary 5.

For any fixed integer

n \geq 2

, let

A

be a

C^{*}

-algebra, U be a square closed Lie ideal of

A

. If

A

admits a nonzero symmetric linear n-derivation

D : A^{n} \to A

with trace

d : A \to A

satisfying one of the following conditions:

(i): $d ([x, y]) \pm d (x) \pm [x, y] \in Z (A) f o r a l l x, y \in U$
(ii): $d ([x, y]) \pm d (y) \pm [x, y] \in Z (A) f o r a l l x, y \in U$

Then

U \subseteq Z (A)

Theorem 7.

For any fixed integer

n \geq 2

, let

A

be a

C^{*}

-algebra, U be a square closed Lie ideal of

A

. If

A

admits a nonzero symmetric linear generalized n-derivation

G : A^{n} \to A

with trace

g : A \to A

associated with symmetric linear n-derivation

D : A^{n} \to A

with trace

d : A \to A

satisfying one of the following conditions:

(i): $g (x) \circ g (y) \pm x \circ y \in Z (A) f o r a l l x, y \in U$
(ii): $g (x) \circ g (y) \pm [x, y] \in Z (A) f o r a l l x, y \in U$

Then

U \subseteq Z (A)

Proof. (i) Suppose on the contrary that

U ⊈ Z (A)

. It is given that

g (x) \circ g (y) \pm x \circ y \in Z (A) f o r a l l x, y \in U .

Replacing y by

y + m z

, where

z \in U

and

1 \leq m \leq n - 1

in the given condition, we get

g (x) \circ g (y + m z) \pm x \circ (y + m z) \in Z (A) f o r a l l x, y, z \in U

which on solving, we have

\begin{matrix} g (x) \circ g (y) + g (x) \circ g (m z) + g (x) \circ {\sum^{n}}_{t = 1}^{n - 1} C_{t} G (\underset{(n - t) - times}{\underset{︸}{y, \dots, y}}, \underset{t - times}{\underset{︸}{m z, \dots, m z}}) \\ \pm x \circ y \pm x \circ m z \in Z (A) f o r a l l x, y, z \in U . \end{matrix}

By using hypothesis, we get

g (x) \circ {\sum^{n}}_{t = 1}^{n - 1} C_{t} D (\underset{(n - t) - times}{\underset{︸}{y, \dots, y}}, \underset{t - times}{\underset{︸}{m z, \dots, m z}}) \in Z (A) f o r a l l x, y, z \in U

which implies that

m A_{1} (x, y, z) + m^{2} A_{2} (x, y, z) + \dots + m^{n - 1} A_{n - 1} (x, y, z) \in Z (A)

for all

x, y, z \in U

where

A_{t} (x, y, z)

represents the term in which z appears t-times.

Making use of Lemma 3, we see that

g (x) \circ G (y, \dots, y, z) \in Z (A) f o r a l l x, y, z \in U .

In particular,

z = y

, we get

g (x) \circ g (y) \in Z (A) f o r a l l x, y \in U .

Hence, by using the given condition, we find that

x \circ y \in Z (A)

for all

x, y \in U

. Replacing x by

y x

, we get

y (x \circ y) \in Z (A)

for all

x, y \in U

. We can also write it as

[y (x \circ y), z] f o r a l l x, y, z \in U

which on solving, we get

[y, z] x \circ y = 0

for all

x, y, z \in U

. Again replace x by

x z

and using the same equation, we get

[y, z] x [z, y] = 0

for all

x, y, z \in U

. Using Lemma 1, we have

[z, y] = 0

for all

z, y \in U

. By Lemma 2, we have

U \subseteq Z (A)

which is a contradiction.

(i i)

Proceeding in the same way as in

(i)

, we conclude. □

Corollary 6.

For any fixed integer

n \geq 2

, let

A

be a

C^{*}

-algebra, U be a square closed Lie ideal of

A

. If

A

admits a nonzero symmetric linear n-derivation

D : A^{n} \to A

with trace

d : A \to A

satisfying one of the following conditions:

(i): $d (x) \circ d (y) \pm x \circ y = 0 f o r a l l x, y \in U$
(ii): $d (x) \circ d (y) \pm [x, y] = 0 f o r a l l x, y \in U$

then

U \subseteq Z (A)

Author Contributions

All authors made equal contributions.

Funding

This research is funded by Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Data Availability Statement

All data required for this paper are included within this paper.

Acknowledgments

The authors extend their appreciation to Princess Nourah Bint Abdulrahman University (PNU), Riyadh, Saudi Arabia for funding this research under Researchers Supporting Project Number (PNURSP2024R231).

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Linear Generalized N-derivations on C-algebras*

Abstract

2. Introduction

3. The Results

Author Contributions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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