1. Introduction
A Banach algebra is a linear associate algebra which, as a vector space, is a Banach space with norm satisfying the multiplicative inequality; ∀ and in . “An involution on an algebra is a linear map of into itself such that the following conditions hold: , , and ∀ and , the field of complex numbers, where is the conjugate of . An algebra equipped with an involution ∗ is called a ∗-algebra or algebra with involution. A Banach ∗-algebra is a Banach algebra together with an isometric involution ∀. A -algebra is a Banach ∗-algebra with the additional norm condition ∀.
Throughout this discussion, unless otherwise mentioned, will denote -algebra with as its center. However, may or may not have unity. The symbols and denote the commutator and the anti-commutator , respectively, for any . An algebra is said to be prime if implies that either or , and semiprime if implies that , where . An additive subgroup U of is said to be a Lie ideal of if , ∀, . U is called a square-closed Lie ideal of if U is a Lie ideal and ∀. A linear operator on a -algebra is called a derivation if holds ∀. Consider the inner derivation implemented by an element a in , which is defined as for every in , 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
is said to be a bi-derivation if
holds for any
. The foregoing conditions are identical if
is also a symmetric map, whereby if
for every
. In this case,
is referred to as a symmetric bi-derivation of
. 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
-algebras. The most general and important one among them is the notion of symmetric linear generalized
n-derivations on
-algebras. Suppose
n is a fixed positive integer and
. A map
is said to be symmetric (permuting) if the relation
holds ∀
,
and for every permutation
. The concept of derivation and symmetric bi-derivation was generalized by Park [
5] as follows: a
n-linear map
is said to be a symmetric (permuting) linear
n-derivation if
is permuting and
hold ∀
. A map
defined by
is called the trace of
. If
is permuting and
n-linear, then the trace
of
satisfies the relation
∀
, where
and
Ashraf et al. [
6] introduced the notion of symmetric generalized
n-derivations in a ring, building upon the concept of generalized derivation. Let
be a fixed positive integer. A symmetric
n-linear map
is known to be symmetric linear generalized
n-derivation if there exists a symmetric linear
n-derivation
such that
holds ∀
.
Example 1. Letwhere is a complex field. Next, define an involution ∗ to be the identity map. It is clear that is a -algebra under norm defined by for all . Denote , , , and let us define by with trace define by . Then it is easy to see that is a symmetric linear generalized n-derivation on . There has been notable scholarly focus on the structure of linear derivations and linear bi-derivations within the context of
-algebras. Various authors have provided diverse expositions of derivations on
-algebras, showcasing a spectrum of perspectives and methodologies. For instance, Kadison’s work in 1966 [
7] demonstrated that every linear derivation acting on a
-algebra annihilates its center. In 1989, Mathieu [
8] built upon Posner’s first theorem [
9] regarding
-algebras, extending its implications. Basically, he proved that “if the product of two linear derivations
d and
on a
-algebra is a linear derivation then
”. Very recently, Ekrami and Mirzavaziri [
10] showed that “if
is a
-algebra admitting two linear derivations
d and
on
, then there exists a linear derivation
D on
such that
if and only if
d and
are linearly dependent”.
In [
11], Ali and Khan proved that if
is a
-algebra admitting a symmetric bilinear generalized ∗-biderivation
with an associated symmetric bilinear ∗-biderivation
, then
maps
into
. 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
and
U be a square closed Lie ideal of
R. Suppose that
is a symmetric bi-derivation and
f, the trace of
B. If
∀
, then either
or
” (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
-algebras and prime rings, respectively. In this study, we extend the above mentioned inquiry to the realm of linear generalized
n-derivations in
-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
-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
-algebras. Precisely, we prove that if
is a
-algebra,
U is a square closed Lie ideal of
admitting a nonzero symmetric linear generalized
n-derivation
with trace
associated with symmetric linear
n-derivation
with trace
satisfying the condition
∀
, then
.
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 and .- 1.
If , then .
- 2.
If (), then .
- 3.
If U is a square closed Lie ideal and , then and .
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 , then . Lemma 3 ([
22])
. Let n be a fixed positive integer and R a -torsion free ring. Suppose that satisfy for . Then for ”. Daif and Bell [
23] proved that if a semiprime ring admits a derivation
d such that either
or
holds ∀
, 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
.
Theorem 1. For any fixed integer , let be a -algebra, U be a square closed Lie ideal of . If admits a nonzero symmetric linear generalized n-derivation with trace associated with symmetric linear n-derivation with trace satisfying the condition of ∀, then .
Proof. It is given that
Replacing
by
, where
in the given condition, we obtain
which on solving, we have
By using hypothesis, we obtain
∀
. Making use of Lemma 3, we see that
For
, (
1) can also be written as
Again making use of Lemma 3, we have
From (
2) and (
3), we obtain
∀
. As every
-algebra is a semiprime ring, using Lemma 2, we obtain
. □
Theorem 2. For any fixed integer , let be a -algebra and U be a square closed Lie ideal of . If admits a nonzero symmetric linear generalized n-derivation with trace associated with symmetric linear n-derivation with trace satisfying the condition of ∀, then .
Proof. Suppose on the contrary that
. We have given that
Replacing
by
, where
and
in the given condition, we obtain
which on solving, we have
Using the given condition, we obtain
Multiply the above equation by
m which implies that
∀
where
represents the term in which
z appears
l-times.
Making use of Lemma 3, we see that
Replace
by
, we obtain
From hypothesis, we have
∀
. Again replace
by
, we have
which imply
. On solving, we obtain
∀
. Again replace
by
, we have
∀
. By Lemma 1, we have
∀
. Again using Lemma 2, we obtain
, which is a contradiction. □
Theorem 3. Let be a -algebra and U be a square closed Lie ideal of . If admits a nonzero symmetric linear generalized n-derivation with trace associated with symmetric linear n-derivation with trace satisfying ∀, then .
Proof. Suppose on the contrary that
. We have given that
be symmetric linear generalized
n-derivations associated with
of a
-algebra
such that
∀
. Therefore,
is semiprime as
is a
-algebra. Now replacing
by
,
for
in the given condition, we obtain
Further solving, we have
In accordance of the given condition and Lemma 3, we obtain
Replacing
by
, we find that
or
This implies that
∀
. Replacing
by
, where
, we obtain
. Again replacing
z by
, we obtain
∀
. Using the Lemma 1, we obtain
∀
. By Lemma 2, we obtain
, a contradiction. □
Corollary 1. For any fixed integer , let be a -algebra and U be a square closed Lie ideal of . If admits a nonzero symmetric linear generalized n-derivation with trace associated with symmetric linear n-derivation with trace satisfying ∀, then .
Theorem 4. For any fixed integer , let be a -algebra and U be a square closed Lie ideal of . Let admit a nonzero symmetric linear generalized n-derivation with trace associated with symmetric linear n-derivation with trace satisfying one of the following conditions:
- (i)
- (ii)
Then, .
Proof. Given that
Consider a positive integer
m;
. Replacing
by
, where
in (
5), we obtain
On further solving, we obtain
On taking account of hypothesis, we see that
where
represents the term in which
z appears
l-times.
Using Lemma 3, we have
In particular, for
, we obtain
Now using the given condition, we find that
From Lemma 2,
.
Follows from the first implication with a slight modification. □
Corollary 2. For any fixed integer , let be a -algebra and U be a square closed Lie ideal of . Let admit a nonzero symmetric linear generalized n-derivation with trace associated with symmetric linear n-derivation with trace satisfying one of the following conditions:
- (i)
- (ii)
Then, .
Corollary 3. For any fixed integer , let be a -algebra and U be a square closed Lie ideal of . Let admit a nonzero symmetric linear generalized n-derivation with trace associated with symmetric linear n-derivation with trace satisfying one of the following conditions:
- (i)
- (ii)
Then, .
Theorem 5. For any fixed integer , let be a -algebra and U be a square closed Lie ideal of . If admits a nonzero symmetric linear generalized n-derivation with trace associated with symmetric linear n-derivation with trace satisfying the condition , then .
Proof. Replacing
by
for
,
in the given condition, we obtain
On further solving and using the specified condition, we obtain
which implies that
∀
where
represents the term in which
z appears
l-times. Using Lemma 3, we obtain
For
, we obtain
then our hypothesis reduces to
. Using the Lemma 2, we obtain
. □
Corollary 4. For any fixed integer , let be a -algebra and U be a square closed Lie ideal of . If admits a nonzero symmetric linear generalized n-derivation with trace associated with symmetric linear n-derivation with trace satisfying the condition , then .
Theorem 6. For any fixed integer , let be a -algebra and U be a square closed Lie ideal of . Let admit a nonzero symmetric linear generalized n-derivation with trace associated with symmetric linear n-derivation with trace satisfying one of the following conditions:
- (i)
- (ii)
Then, .
Proof. Given that
Replacing
by
, where
and
in the given condition, we obtain
which on solving and using hypothesis, we obtain
which implies that
∀
where
represents the term in which
z appears
l-times.
Making use of Lemma 3 and torsion restriction, we see that
Replace
z by
to obtain
Hence, by using the given condition, we find that
. On taking account of Lemma 2, we obtain
.
Given that
Replacing
by
, where
and
in the given condition, we obtain
which on solving and using hypothesis, we obtain
which implies that
∀
where
represents the term in which
z appears
l-times.
Making use of Lemma 3 and torsion restriction, we see that
Replace
z by
to obtain
Hence, by using the given condition, we find that
. On taking account of Lemma 2, we obtain
.
Follows from the first implication with a slight modification. □
Corollary 5. For any fixed integer , let be a -algebra and U be a square closed Lie ideal of . Let admit a nonzero symmetric linear n-derivation with trace satisfying one of the following conditions:
- (i)
- (ii)
Then, .
Theorem 7. For any fixed integer , let be a -algebra and U be a square closed Lie ideal of . Let admit a nonzero symmetric linear generalized n-derivation with trace associated with symmetric linear n-derivation with trace satisfying one of the following conditions:
- (i)
- (ii)
Then, .
Proof. Suppose on the contrary that
. It is given that
Replacing
by
, where
and
in the given condition, we obtain
which on solving, we have
By using hypothesis, we obtain
which implies that
∀
where
represents the term in which
z appears
l-times.
Making use of Lemma 3, we see that
In particular,
, we obtain
Hence, by using the given condition, we find that
∀
. Replacing
by
, we obtain
∀
. We can also write it as
which on solving, we obtain
∀
. Again replace
by
and using the same equation, we obtain
∀
. Using Lemma 1, we have
∀
. By Lemma 2, we have
which is a contradiction.
Proceeding in the same way as in , we conclude. □
Corollary 6. For any fixed integer , let be a -algebra and U be a square closed Lie ideal of . Let admit a nonzero symmetric linear n-derivation with trace satisfying one of the following conditions:
- (i)
- (ii)
Then, .