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Jordan derivations and Jordan left derivations of Banach algebras
Abstract
We show that every left derivation on a semiprime Banach algebra A is a derivation which maps A into the intersection of the center of A and the Jacobson radical of A, and hence every left derivation on a semisimple Banach algebra is always zero.
... In this direction the range of left derivations was reviewed. In the year 1998, Jung [99] proved that every spectrally bounded left derivation maps algebra into its Jacobson radical. Vukman [161] showed that every Jordan left derivation on a semisimple Banach algebra is identically zero. ...
... In 1992, Vukman [157] proved the result for semiprime rings. He was able to accomplish the following; In 1998, Jung [99] investigated the above study for semiprime Banach algebra and was able to establish; Theorem 3.6. Every left derivation on a semiprime Banach algebra A is a derivation which maps A into the intersection of the center of A and the Jacobson radical of A. This is evident that every left derivation forms a Jordan left derivation but the converse need not be true in general. ...
In this overview article, we provide a historical account on derivations, Jordan derivations, (α, β)-derivations, left derivations, pre-derivations, homoderivations, nilpotent derivations, and other variants, drawing from the contributions of multiple researchers. Additionally, we delve into recent findings and suggest potential avenues for future investigation in this area. Furthermore, we offer pertinent examples to illustrate that the assumptions underlying various results are indeed necessary and not redundant.
... The (Jordan) left derivation has been widely studied in some special rings and algebras (see [3][4][5][6][7]). Concretely, related results have been proved in prime rings (see [8]), semi-prime rings (see [9]), and Γ-rings (see [10,11]); on the other hand, related results have been proved in Banach algebras (see [12][13][14][15][16][17]). Some results were generalized to different types of semi-rings, which play an important role in theoretical computer science (see [18,19]). ...
Let R R be a unital associative ring. Our motivation is to prove that left derivations in column finite matrix rings over R R are equal to zero and demonstrate that a left derivation d : T → T d:{\mathcal{T}}\to {\mathcal{T}} in the infinite upper triangular matrix ring T {\mathcal{T}} is determined by left derivations d j {d}_{j} in R ( j = 1 , 2 , … ) R\left(j=1,2,\ldots ) satisfying d ( ( a i j ) ) = ( b i j ) d\left(\left({a}_{ij}))=\left({b}_{ij}) for any ( a i j ) ∈ T \left({a}_{ij})\in {\mathcal{T}} , where b i j = d j ( a 11 ) , i = 1 , 0 , i ≠ 1 . {b}_{ij}=\left\{\begin{array}{ll}{d}_{j}\left({a}_{11}),& i=1,\\ 0,& i\ne 1.\end{array}\right. The similar results about Jordan left derivations are also obtained when R R is 2-torsion free.
... In the following, some significant works on the range of left derivations are reviewed. In 1998, Jung [10] proved that every spectrally bounded left derivation maps algebra into its Jacobson radical. Vukman [19] showed that every Jordan left derivation on a semisimple Banach algebra is identically zero. ...
Let A be an algebra, and let I be a semiprime ideal of A. Suppose thatd : A → A is a Jordan left derivation such that d(I) ⊆ I.We prove that if dim{d(a)+I : a ⋲ A} ≤ 1, then d(A) ⊆ I. Additionally, we consider several consequences of this result.
... Results concerning Jordan left derivations can be found in [7], [23], [24], [31], [35], [40], [41], [42], [50], [61], [71], [72], [74], [79] ...
In this brief survey we consider the Jordan derivations on rings and semirings.
... A linear mapping d : A → A is called a derivation if d(ab) = d(a)b + ad(b) holds for all pairs a, b ∈ A and is called a Jordan derivation in case d(a 2 ) = d(a)a + ad(a) is fulfilled for all a ∈ A. A left derivation on A is a linear mapping L : A → A if L(ab) = aL(b) + bL(a) holds for all pairs a, b ∈ A and is called a Jordan left derivation if L(a 2 ) = 2aL(a) is fulfilled for all a ∈ A. Recently, a number of authors ( [1,8,13,21,23]) have studied left derivations and various generalized notions of them in the context of pure algebra, extensively. As a pioneering work, Bresar and Vukman [5] proved that every left derivation on a semiprime ring R is a derivation which maps R into its center. ...
In this study, we show that every continuous Jordan left derivation on a (commutative or noncommutative) prime UMV-Banach algebra with the identity element 1 is identically zero. Moreover, we prove that every continuous left derivation on a unital finite dimensional Banach algebra, under certain conditions, is identically zero. As another result in this regard, it is proved that if ℜ is a 2-torsion free semiprime ring such that ann{[y, z] | y, z ∈ ℜ} = {0}, then every Jordan left derivation L: ℜ → ℜ is identically zero. In addition, we provide several other results in this regard.
Let G be a simple graph of order n and A(G) be an adjacency matrix of G, the spectra of G is the set of eigenvalues of A(G). In other side, the resolvent matrix is a matrix with property that all of its eigenvalues are outside the spectra. In this talk, we present an exponential growth to the graph resolvent.
In this paper, we introduced the concepts of semi global domination number in product fuzzy graph, and is denoted by γsg(G) and semi complimentary product fuzzy graph. We determine the semi global domination number γsg(G) for several classes of product fuzzy graph and obtain Nordhaus-Gaddum type results for this parameter. Further, some bounds of γsg(G) are investigated. Moreover, the relations between γsg(G) and other known parameters in Product fuzzy graphs are investigated.
In this part, we will introduce definitions of special functions. Gamma, beta and Mittag- Leffler functions will be given. Then, derivatives and integration of Mittag-Leffler function will be defined before definitions of fractional operators will have been mentioned. Since, this topic are basic to introduce q-Mittag Leffler function and their applications on q-fractional differential operators. Finally, q-gamma, q-beta and q-Mittag Leffler functions will be mentioned before q-fractional differential operators.
In this section, initially we will mention that the definitions of fractional calculus. To give a detail as, Grunwald-Letnikov, Riemann-Liouville & Caputo's definitions will be given because, these definitions are connected with the difference operators. So, after these definitions are introduced, difference equations and linear type of difference equations will be given in this part. Later, definitions of fractional calculus that are our mentioning will be adapted in q-version as Riemann-Liouville and Caputo type on difference operators.
The purpose of this paper is to obtain Jordan left derivations that map into the Jacobson radical: (i) Let d be a spectrally bounded Jordan left derivation on a Banach algebra A. If [d(x),x]∈rad(A) for all x∈A, then d(A)⊆rad(A). (ii) Let d be a Jordan left derivation on a unital Banach algebra A with the condition sup{r(z -1 d(z))∣z∈Ainvertible}<∞. If [d(x),x]∈rad(A) for all x∈A, then d(A)⊆rad(A).
We intend to provide an overview of some recent work on a problem on unbounded derivations of Banach algebras that still defies solution, the non-commutative Singer-Wermer conjecture. In particular, we discuss several global as well as local properties of derivations entailing quasinilpotency in the image.
In 1955 I. M. Singer and J. Wermer proved that a bounded derivation on a commutative Banach algebra maps into the (Jacobson) radical; they conjectured that this result holds even if the derivation is unbounded. We give a proof of this conjecture. The central idea in the proof is the introduction of the concept of a recalcitrant system of elements in a commutative radical Banach algebra. Such systems put algebraic constraints upon a derivation which prevent the derivation from mapping outside of the radical.
Let R be a ring and X be a left R-module. The purpose of this paper is to investigate additive mappings D1: R → X and D2: R → X that satisfy D1(ab) = aD1(b) + bD1(a), a, b ∈ R (left derivation) and D2(a2) = 2aD2(a), a ∈ R (Jordan left derivation). We show, by the rather weak assumptions, that the existence of a nonzero Jordan left derivation of R into X implies R is commutative. This result is used to prove two noncommutative extensions of the classical Singer-Wermer theorem.
I. N. Herstein has proved that any Jordan derivation on a 2-torsion free prime ring is a derivation. In this paper we prove that Herstein’s result is true in 2-torsion free semiprime rings. This result makes it possible for us to prove that any linear Jordan derivation on a semisimple Banach algebra is continuous, which gives an affirmative answer to the question posed by A. M. Sinclair [in Proc. Am. Math. Soc. 24, 209-214 (1970; Zbl 0175.440)].
