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Abstract
A map is called a strong commutativity preserving (SCP) on
if
for all
, where
is a Banach algebra over
or
and
is a nonempty subset of
. The main goal of this manuscript is to investigate the SCP skew derivations on Banach algebra
and we obtain action of skew derivations on rings and the structure of prime Banach algebras.
1. Introduction
In this manuscript, we investigate strong commutativity preserving (SCP) skew derivations on prime Banach algebra. The concept of SCP derivations is first studied by Bell and Daif [Citation1]. From then onward many researchers have discussed the behaviour of SCP maps on different subsets of rings and established the structure of rings/algebras. Our main motive is to study the SCP maps on Banach algebras, instead of using ring theoretic computations, our approach is somewhat different and based on the theory of topology and analysis.
Throughout the paper represents a prime Banach algebra. A linear map d of
into itself is called a linear derivation if
for all
. Let ζ be an automorphism of
. A linear ζ-derivation
is a linear map satisfying
It is generally called a linear skew derivation. When
on
linear ζ-derivation is simply an ordinary linear derivation. For
, the example of linear ζ-derivation is the map
, where
denotes the identity automorphism of
. Thus the results on linear skew derivations are the generalizations of both linear derivations and automorphisms. Moreover, linear skew derivation is a non-identity endomorphism on
.
Numerous results in the literature specify how the general structure of rings and algebras are mostly connected to the behaviour of linear maps on rings and algebras [Citation2–9]. In [Citation1], Bell and Daif explored the commutativity of a ring if it satisfies SCP condition on a right ideal of
. Later, Brešar and Miers [Citation10] studied additive SCP maps on semiprime rings and characterized them. In particular, they obtained: Let
be an additive map on a semiprime ring
satisfying SCP on
, then
, where
, the extended centroid of
,
, and ν is an additive map of
to
. Later, Deng and Ashraf [Citation11] established that if there exists a non-identity endomorphism φ on
, where
is a prime ring of characteristic different from two, such that
for each x,y in some right ideal of
, then
is commutative. In 2016, De Filippis et al. [Citation12] discussed the strong commutative preserving skew derivation on ideals of prime rings. More exactly, they proved that: Let
be a prime ring of characteristic different from 2, C be the extended centroid of
,
be the centre of
, I be a nonzero ideal of
, F and G be two nonzero skew derivations of
with associated automorphism α and m,n be the positive integers such that
for all
. Then
is commutative.
In the past few years, numerous algebraists has been investigated a lot of ring theoretic results and try to relate it to Banach algebras by using the standard approach and established some relationship between purely ring theoretic results and Banach algebras. Some of them discussed their results on the theory of Singer–Wermer result [Citation13], who proved that every linear derivation on a Banach algebra maps into its Jacobson radical. In this line of investigation, our aim is to explore SCP skew derivations on Banach algebras. Also our approach is somewhat different and based on the theory of rings, topology and analysis. In particular, we prove the following:
Theorem 1.1
Let δ be a linear continuous skew derivation on into itself. Suppose
are open subsets of
and
depending on x and y such that
for each
. Then
is commutative.
Corollary 1.1
Let d be a linear continuous derivation on into itself. Suppose
are open subsets of
and
depending on x and y such that
for each
. Then
is commutative.
Lastly, in favour of our main theorem, we present the following example:
Example 1.1
Let be a noncommutative unital prime algebra of all
matrices over
, where
is a field of complex numbers, with usual matrix addition, and define matrix multiplication as follows:
where
and
. For
, set the Frobenius norm
on
as follows:
Then,
is a normed linear space under the defined norm. Further, define a map
by
and
for every
Since,
is finite-dimensional, it is straightforward to check that δ is a nonzero continuous linear skew-derivation (ζ-derivation) on
. Observe that
and
are open subsets of
such that
Therefore, it follows from Theorem 1.1 that
is not a Banach algebra under the Frobenius norm
.
2. The main result
Before starting the proof of our main theorem, we need the following crucial proposition which is used in the sequel.
Proposition 2.1
Let be a prime ring with characteristic not two. If there exists a linear skew derivation δ on
into itself satisfying
for all
where
. Then
is commutative unless
and satisfies
the standard identity of degree 4.
Proof.
Let and
be additive subgroups. It implies that
for all
and
. By [Citation14, Main Theorem] either
have a Lie ideal (non-central)
or
. The latter case concludes
is commutative. Similarly, assume that there exists a Lie ideal
of
such that
. Moreover, by [Citation15] (page 4–5), there exist
and
ideals of
, such that
and
. Thus we have
for each
and
. Since
,
and
satisfy the same GPIs by [Citation16, Citation17], so we conclude that
(1)
(1) for each
. Application of [Citation18, Corollary 1.4] yields
, i.e.
is commutative unless
and satisfies
the standard identity of degree 4.
Proof
Proof of Theorem 1.1
Fix and define
{
for each k,l>1. From here, it is easy to see that
is open. If every
is dense, we know that their intersection is also dense by Baire category theorem. This contradicts the existence of
and
. Hence for
,
is not a dense set and there exists an nonempty open set
in the complement of
such that
for all
. Take
,
and
,
we have
(2)
(2) The expression
can be written as
Let
. If
, then
represents the summation of all those terms in which
appears α times, and w appears β times in the expansion of
. Similarly, we can define
for
. The coefficient of
in above polynomial is just
. Hence
. Therefore for
there are
depending on x and y such that for each
,
Now, fix and for
, set
{
}. Obviously, each
is open. If each
is dense, then the intersection is also. However, this is contrary to what was shown earlier concerning the open set
. Hence there are integers
and a nonempty open subset
in
. If
and
,
for all small
. Hence, for positive integers
for each
and
. Arguing as above, we see that
for each
Now consider {
for each
, then the union of
will be
It can be easily proven that each
is closed. Hence some
,
, must have a nonempty open subset
, by Baire category theorem. Let
, for all small
and each
Hence by earlier arguments, for each
we have
. Thus by Proposition 2.1, we get the desire conclusion. This completes the proof.
In conclusion, it is tempting to conjecture as follows:
Conjecture 2.1
Let δ be a continuous skew derivation (linear) on into itself, where
is a semisimple Banach algebra. Suppose
,
are open subsets of
and
depending on x and y such that
for each
,
. Then
is commutative.
Disclosure statement
No potential conflict of interest was reported by the authors.
ORCID
Abdul Nadim Khan http://orcid.org/0000-0002-5861-6137
Mohd Arif Raza http://orcid.org/0000-0001-6799-8969
Husain Alhazmi http://orcid.org/0000-0001-7190-5884
References
- Bell HE, Daif MN. On commutativity and strong commutativity preserving maps. Canad Math Bull. 1994;37(4):443–447. doi: 10.4153/CMB-1994-064-x
- Raza MA, Rehman N. On generalized derivation in rings and Banach algebras. Kragujevac J Math. 2017;41(1): 105–120. doi: 10.5937/KgJMath1701105R
- Ali S, Khan AN. On commutativity of Banach algebras with derivations. Bull Aust Math Soc. 2015;91:419–425. doi: 10.1017/S0004972715000118
- Ali S, Khan MS, Khan AN, et al. On rings and algebras with derivations. J Algebra Appl. 2016;15(6):1650107 (10 pages). doi: 10.1142/S0219498816501073
- Raza MA, Rehman N. On prime and semiprime rings with generalized derivations and non-commutative Banach algebras. Proc Indian Acad Sci Math Sci. 2016;126(3): 389–398. doi: 10.1007/s12044-016-0287-2
- Ashraf M, Rehman N, Raza MA. A note on commutativity of semiprime Banach algebras. Beitr Algebra Geom. 2016;57(3):553–560. doi: 10.1007/s13366-015-0264-4
- Raza MA, Khan MS, Rehman N. Some differential identities on prime and semiprime rings and Banach algebras. Rend Circ Mat Palermo (2). 2018. Available from: https://doi.org/10.1007/s12215-018-0358-6.
- Rehman N, Raza MA. On Lie ideals with generalized derivations and non-commutative Banach algebras. Bull Malays Math Sci Soc. 2017;40(2):747–764. doi: 10.1007/s40840-017-0453-4
- Rehman N, Khan MS. A note on multiplicative (generalized)-skew derivation on semiprime rings. J Taibah Univ Sci. 2018;12(4):450–454. doi: 10.1080/16583655.2018.1490049
- Brešar M, Miers CR. Strong commutativity preserving mappings of semiprime rings. Canad Math Bull. 1994;37:457–460. doi: 10.4153/CMB-1994-066-4
- Deng Q, Ashraf M. On strong commutativity preserving maps. Results Math. 1996;30:259–263. doi: 10.1007/BF03322194
- De Filippis V, Rehman N, Raza MA. Strong commutativity preserving skew derivations in semiprime rings. Bull Malays Math Sci Soc. 2018;41(4):1819–1834. doi: 10.1007/s40840-016-0429-9
- Singer IM, Wermer J. Derivations on commutative normed algebras. Math Ann. 1955;129:260–264. doi: 10.1007/BF01362370
- Chuang CL. The additive subgroup generated by a polynomial. Israel J Math. 1987;59(1):98–106. doi: 10.1007/BF02779669
- Herstein IN. Topics in ring theory. Chicago: The University of Chicago Press; 1969.
- Chuang CL. GPIs having coefficients in Utumi quotient rings. Proc Amer Math Soc. 1988;103:723–728. doi: 10.1090/S0002-9939-1988-0947646-4
- Chuang CL. Differential identities with automorphsims and antiautomorphisms II. J Algebra. 1993;160:130–171. doi: 10.1006/jabr.1993.1181
- Lin JS, Liu CK. Strong commutativity preserving maps on Lie ideals. Linear Algebra Appl. 2008;428:1601–1609. doi: 10.1016/j.laa.2007.10.006