Derivations In Banach Algebras Research Articles

When is a ϕ-derivation continuous and where can its image be found?

This paper has two main objectives. One of them is to obtain some conditions under which the image of a [Formula: see text]-derivation on a Banach algebra [Formula: see text] is contained in the Jacobson radical of [Formula: see text] and other purpose of this study is to present some results about the automatic continuity of [Formula: see text]-derivations in Banach algebras.

On b-generalized skew derivations in Banach algebras

On b-generalized skew derivations in Banach algebras

Bihom derivations in Banach algebras

In this paper, we introduce bihom derivations in complex Banach algebras. Using the fixed point method and the direct method, we prove the Hyers–Ulam stability of bihom derivations in complex Banach algebras, associated with the bi-additive s-functional inequality $$\Vert f(x+y, z-w) + f(x-y, z+w) -2f(x,z)+2 f(y, w)\Vert \le \Vert s \left( 2f\left( \frac{x+y}{2}, z-w\right) + 2f\left( \frac{x-y}{2}, z+w\right) - 2f(x,z )+ 2 f(y, w)\right) \Vert $$ , where s is a fixed nonzero complex number with $$|s |< 1$$ .

Point derivations and cohomologies of Lipschitz algebras

AbstractFor a compact metric space (K, d), LipK denotes the Banach algebra of all complex-valued Lipschitz functions on (K, d). We show that the continuous Hochschild cohomology Hn(LipK, (LipK)*) and Hn(LipK, ℂe) are both infinite-dimensional vector spaces for each n ≥ 1 if the space K contains a certain infinite sequence which converges to a point e ∈ K. Here (LipK)* is the dual module of LipK and ℂe denotes the complex numbers with a LipK-bimodule structure defined by evaluations of LipK-functions at e. Examples of such metric spaces include all compact Riemannian manifolds, compact geodesic metric spaces and infinite compact subsets of ℝ. In particular, the (small) global homological dimension of LipK is infinite for every such space. Our proof uses the description of point derivations by Sherbert [‘The structure of ideals and point derivations in Banach algebras of Lipschitz functions’, Trans. Amer. Math. Soc.111 (1964), 240–272] and directly constructs non-trivial cocycles with the help of alternating cocycles of Johnson [‘Higher-dimensional weak amenability’, Studia Math.123 (1997), 117–134]. An alternating construction of cocycles on the basis of the idea of Kleshchev [‘Homological dimension of Banach algebras of smooth functions is equal to infinity’, Vest. Math. Mosk. Univ. Ser. 1. Mat. Mech.6 (1988), 57–60] is also discussed.

On skew derivations and generalized skew derivations in Banach algebras

Let A be unital prime Banach algebra over ℝ or ℂ with centre and G 1, G 2 be open subsets of be a continuous linear generalized skew derivation, and be a continuous linear map. We prove that must be commutative if one of the following conditions holds: For each a ∈ G 1 , b ∈ G 2, there exists an integer m ∈ Z >1 depending on a and b such that either . For each a ∈ G 1 , b ∈ G 2, there exists an integer m ∈ Z >1 depending on a and b such that either . These results generalize a number of theorems of this type. In particular, as an application, we give an affirmative answer to some questions posed in [21].

Asymptotic aspect of derivations in Banach algebras

We prove that every approximate linear left derivation on a semisimple Banach algebra is continuous. Also, we consider linear derivations on Banach algebras and we first study the conditions for a linear derivation on a Banach algebra. Then we examine the functional inequalities related to a linear derivation and their stability. We finally take central linear derivations with radical ranges on semiprime Banach algebras and a continuous linear generalized left derivation on a semisimple Banach algebra.

Skew Derivations in Banach Algebras

AbstractWe investigate the global versions of the Kleinecke–Shirokov theorem for skew derivations in Banach algebras. Centralizing skew derivations on Banach algebras are also studied.

ON APPROXIMATE n-ARY DERIVATIONS

C. Park et al. proved the stability of homomorphisms and derivations in Banach algebras, Banach ternary algebras, C*-algebras, Lie C*-algebras and C*-ternary algebras. In this paper, we improve and generalize some results concerning derivations. We first introduce the following generalized Jensen functional equation [Formula: see text] and using fixed point methods, we prove the stability of n-ary derivations and n-ary Jordan derivations in n-ary Banach algebras. Secondly, we study this functional equation with *-n-ary derivations in C*-n-ary algebras.

A note on generalized derivations in Banach algebras

In this note we characterize a complex Banach algebra A admitting a generalized derivation g such that the cardinality of the spectrum σ ( g ( x ) ) is exactly one for all x ∈ A .

On the composition of [formula omitted]-skew derivations in Banach algebras

Let A be a Banach algebra, let σ be an automorphism of A , and let d, δ be q-skew σ-derivations of A . We show that if d δ ( a ) is quasi-nilpotent for any a ∈ A , then d δ ( a ) 3 lies in the radical of A for all a ∈ A . This result simultaneously generalizes the previous results obtained by Brešar and Šemrl [5], Chebotar et al. [7] and Abdelali [1].

On generalized derivations in Banach algebras

We study generalized derivations $G$ defined on a complex Banach algebra $A$ such that the spectrum $\sigma (Gx)$ is finite for all $x \in A$. In particular, we show that if $A$ is unital and semisimple, then $G$ is inner and implemented by elements of th

ON THE STABILITY OF MAPPINGS IN BANACH ALGEBRAS

We obtain the stability of additive, multiplicative mappins and derivations in Banach algebras.

Derivations in Banach algebras

We present some conditions which imply that a derivation on a Banach algebra maps the algebra into its Jacobson radical.

A Result Concerning Derivations in Banach Algebras

A Result Concerning Derivations in Banach Algebras

A result concerning derivations in Banach algebras

The main result: Let A A be a Banach algebra over the complex field C C . Suppose there exists a continuous derivation D : A → A D:A \to A , such that α D 3 + D 2 \alpha {D^3} + {D^2} is a derivation for some α ∈ C \alpha \in C . In this case D D maps A A into its radical.

Some remarks on derivations in Banach algebras and related results

In the first section of this paper we consider some functional equations which are closely connected to derivations (i.e. additive mappings with the propertyD(ab) = aD(b) + D(a)b) on Banach algebras. IfD is a derivation on some algebraA, then the equationD(a) = − aD(a−1)a holds for all invertible elementsa ∈A. It seems natural to ask whether this functional equation characterizes derivations among all additive mappings. It is too much to expect an affirmative answer to this question in arbitrary algebras, since it may happen that even in normed algebras the group of all invertible elements contains only scalar multiples of the identity. We try to answer the question above in Banach algebras, since in Banach algebras invertible elements exist in abundance. In the second section of the paper we prove some results concerning representability of quadratic forms by bilinear forms.

The Structure of Ideals and Point Derivations in Banach Algebras of Lipschitz Functions

The Structure of Ideals and Point Derivations in Banach Algebras of Lipschitz Functions

The structure of ideals and point derivations in Banach algebras of Lipschitz functions

Introduction. The main object of study in this paper is the space of all bounded complex-valued functions defined on a metric space (X, d) which satisfy a Lipschitz condition with respect to the metric d. With a suitable norm this collection of functions becomes a Banach algebra. This Banach algebra, which we shall call a Lipschitz algebra, will be denoted by Lip(X, d). It is the structure of Lip(X, d) with which we are primarily concerned here. Although the notion of Lipschitz continuity is very old and Lipschitz functions have been studied for many years, interest in the Banach space and Banach algebra theory of Lipschitz functions has not developed until quite recently. Very little is known about the Banach space properties of Lip (X, d). The following papers constitute all the work known to this writer which has been done in this area. Mirkil [9] and de Leeuw [4] have considered the space of periodic Lipschitz functions on the real line (or in other words, Lipschitz functions defined on the circle group). Mirkil was interested in the relation between the Lipschitz condition and the translation properties of functions, the fact that the functions were defined on a group being vital. de Leeuw was more directly concerned with the Banach space structure, his paper involving the study of certain dual spaces, extreme points, and isometries. In the long paper of Glaeser [6], a short chapter, somewhat off the main theme of his work, is devoted to the space of n-times continuously differentiable functions defined on a subset of Emn whose nth derivatives satisfy a Lipschitz condition. It is shown as a side result in the paper of Arens and Eells [1] that Lip(X,d) is always the dual space of some normed linear space. The only work known to us which treats Lip(X,d) as a Banach algebra was done by S. B. Myers [11]. His paper [11] is a summary of results, the proofs never published because of his untimely death in 1955. His interest in Lipschitz algebras was connected with the process of differentiation, and this interest we pursue in Chapter III of the present study. We supply proofs of many of the unproved statements of Myers, and in some cases extend his results to more general settings.