Let ${\mathcal A}$ be a Banach algebra with the properties that $\mathrm{rad}({\mathcal A})={\rm rann}({\mathcal A})$ and the algebra ${\mathcal A}/\mathrm{rad}({\mathcal A})$ is commutative. We show that a derivation of ${\mathcal A}$ maps ${\mathcal A}$ into ${\rm rad}({\mathcal A})$. Using this, we determine among other things when a generalized derivation of ${\mathcal A}$ maps ${\mathcal A}$ into ${\rm rad}({\mathcal A})$. We also study $k$-centralizing generalized derivations of ${\mathcal A}$. Then, for a generalized derivation $(δ, d)$ of ${\mathcal A}$ we obtain a necessary and sufficient condition for $(δ^2, d^2)$ to be still a generalized derivation of ${\mathcal A}$. The main applications are concerned with the algebras over locally compact groups. In particular, we deduce these results for bidual of Fourier algebras of discrete amenable groups as an application of our approach.
In this paper, we study several type of point derivations for Banach algebras. We investigate how our definition of point derivations are related to each others.
In this article, it is proved that a functional equation of (linear) Jordan triple derivations on unital Banach algebras under quite natural and simple assumptions is hyperstable. It is also shown that under some mild conditions approximate Jordan triple derivations on unital semiprime Banach algebras are (linear) derivations.
We consider the compactness of derivations from commutative Banach algebras into their dual modules. We show that if there are no compact derivations from a commutative Banach algebra, $A$, into its dual module, then there are no compact derivations from $A$ into any symmetric $A$-bimodule; we also prove analogous results for weakly compact derivations and for bounded derivations of finite rank. We then characterise the compact derivations from the convolution algebra $\ell^1(\Z_+)$ to its dual. Finally, we give an example (due to J. F. Feinstein) of a non-compact, bounded derivation from a un…
In this article we shall focus on the derivations on module extension of Banach algebras and determine the general structure of them. Then we obtain some results concerning the automatic continuity of these mappings.
In this paper, we study Jordan derivation-like maps on the $θ-$Lau products of algebras. We characterize them and prove that under certain condition any Jordan derivation-like maps on the $θ-$Lau products is a derivation-like map. Moreover, we investigate the concept of centralizing for Jordan derivation-like maps on the $θ-$Lau products of algebras.
Let $A$ be a Banach algebra and $X$ be a Banach $A$-bimodule. A mapping $D :A\longrightarrow X$ is a cubic derivation if $D$ is a cubic homogeneous mapping, that is $D$ is cubic and $D(λa)=λ^3 D(a)$ for any complex number $λ$ and all $a\in A$, and $D(ab)=D(a)\cdot b^3 +a^3\cdot D(b)$ for all $a,b\in A$. In this paper, we prove the stability of a cubic derivation with direct method. We also employ a fixed point method to establish of the stability and the superstability for cubic derivations.
Let $X$ be a left introverted subspace of dual of a Banach algebra. We study $Z_t(X^*),$ the topological center of Banach algebra $X^*$. We fined the topological center of $(X\cA)^*$, when $\cA$ has a bounded right approximate identity and $\cA\subseteq X^*.$ So we introduce a new notation of amenability for a dual Banach algebra $\cal A$. A dual Banach algebra $\cal A$ is weakly Connes-amenable if the first $weak^*-$continuous cohomology group of $\cal A$ with coefficients in $\cal A$ is zero; i.e., $H^1_{w^*}(\cal A, \cal A)=\{o\}$. We study the weak Connes-amenability of some dual Banach al…
Let A and U be Banach algebras such that U is also a Banach A- bimodule with compatible algebra operations, module actions and norm. By defining an approprite action, we turn l1-direct product A item U into a Banach algebra such that A is closed subalgebra and U is a closed ideal of it. This algebra, is in fact semidirect product of A and U which we denote it by A litem U and every semidirect products of Banach algebras can be represented as this form. In this paper we consider the Banach algebra A litem U as mentioned and study the derivations on it. In fact we consider the automatic continui…
We investigate the automatic regularity of bounded derivations from a Banach lattice algebra of regular operators A into a Banach A-module with a Banach lattice structure compatible with the module operations.
Let $A$ be a Banach ternary algebra over a scalar field $\Bbb R$ or $\Bbb C$ and $X$ be a ternary Banach $A-$module. Let $σ,τ$ and $ξ$ be linear mappings on $A$, a linear mapping $D:(A,[]_A)\to (X,[]_X)$ is called a Lie ternary $(σ,τ,ξ)-$derivation, if $$D([abc]_A)=[[D(a)bc]_X]_{(σ,τ,ξ)}+[[D(b)ac]_X]_{(σ,τ,ξ)}+[[D(c)ba]_X]_{(σ,τ,ξ)},$$ for all $a,b,c\in A$, where $[abc]_{(σ,τ,ξ)}=aτ(b)ξ(c)-σ(c)τ(b)a.$ In this paper, we investigate the generalized Hyers--Ulam--Rassias stability of Lie ternary $(σ,τ,ξ)-$derivations on Banach ternary algebras.
Let $\cal A$ be a Banach algebra. We study those closed ideals $I$ of $\cal A$ for which the first cohomology group of $\cal A$ with coefficients in $I^*$ is trivial; i.e. $H^1(\cal A,I^*)=\{0\}$. We investigate such closed ideals when $\cal A$ is weakly amenable or biflat. Also we give some hereditary properties of ideal amenability.
Let $A$ and $X$ be Banach algebras and let $X$ be an algebraic Banach $A-$module. Then the $\ell^1-$direct sum $A\times X$ equipped with the multiplication $$(a,x)(b,y)=(ab, ay+xb+xy)\quad (a,b\in A, x,y\in X)$$ is a Banach algebra, denoted by $A\bowtie X$, which will be called "\textit{a generalized module extension Banach algebra}". Module extension algebras, Lau product and also the direct sum of Banach algebras are the main examples satisfying this framework. We characterize the structure of $n-$dual valued ($n\in\mathbb N)$) derivations on $A\bowtie X$ from which we investigate the $n-$we…
We show that for $n\geq 2$ the $n-$weak amenability of the second dual $\A^{**}$ of a Banach algebra $\A$ implies that of $\A$. We also provide a positive answer for the case $n=1,$ which sharpens some older results. Our method of proof also provides a unified approach to give short proofs for some known results in the case where $n=1$.
In this paper, we introduce and solve the following additive-additive $(s,t)$-functional inequality \begin{eqnarray}\label{0.1} && \|g\left(x+y\right) -g(x) -g(y)\| +\| h(x+y) + h(x-y) -2 h(x) \| && \le \left\|s\left( 2 g\left(\frac{x+y}{2}\right)-g(x)-g(y)\right)\right\|+ \left\|t \left( 2h\left(\frac{x+y}{2}\right)+ 2h \left(\frac{x-y}{2}\right)- 2h (x)\right) \right\| , \nonumber \end{eqnarray} where $s$ and $t$ are fixed nonzero complex numbers with $|s| <1$ and $ |t| <1$. Using the direct method and the fixed point method, we prove the Hyers-Ulam stability of Lie bracket derivation-deriva…
We introduce two notions of amenability for a Banach algebra $\cal A$. Let $I$ be a closed two-sided ideal in $\cal A$, we say $\cal A$ is $I$-weakly amenable if the first cohomology group of $\cal A$ with coefficients in the dual space $I^*$ is zero; i.e., $H^1({\cal A},I^*)=\{0\}$, and, $\cal A$ is ideally amenable if $\cal A$ is $I$-weakly amenable for every closed two-sided ideal $I$ in $\cal A$. We relate these concepts to weak amenability of Banach algebras. We also show that ideal amenability is different from amenability and weak amenability. We study the $I$-weak amenability of a Bana…
If $D:A \to X$ is a derivation from a Banach algebra to a contractive, Banach $A$-bimodule, then one can equip $X^{**}$ with an $A^{**}$-bimodule structure, such that the second transpose $D^{**}: A^{**} \to X^{**}$ is again a derivation. We prove an analogous extension result, where $A^{**}$ is replaced by $\F(A)$, the \emph{enveloping dual Banach algebra} of $A$, and $X^{**}$ by an appropriate kind of universal, enveloping, normal dual bimodule of $X$. Using this, we obtain some new characterizations of Connes-amenability of $\F(A)$. In particular we show that $\F(A)$ is Connes-amenable if a…
Let $\mathcal{A}$ and $\mathcal{U}$ be Banach algebras and $θ$ be a nonzero character on $\mathcal{A}$. Then the \textit{Lau product Banach algebra} $\mathcal{A}\times_θ\mathcal{U}$ associated with the Banach algebras $\mathcal{A}$ and $\mathcal{U}$ is the $l^1$-direct sum $\mathcal{A}\oplus\mathcal{U}$ equipped with the algebra multiplication $(a,u)(a',u')=(ab,θ(a)u'+θ(a')u+uu')\, (a,a'\in\mathcal{A},u,u'\in\mathcal{U})$ and $l^1$-norm. In this paper we shall investigate the derivations and multipliers from this Banach algebras and study the automatic continuity of these mappings. We also stu…
We introduce two notions of amenability for a Banach algebra $\cal A$. Let $n\in \Bbb N$ and let $I$ be a closed two-sided ideal in $\cal A$, $\cal A$ is $n-I-$weakly amenable if the first cohomology group of $\cal A$ with coefficients in the n-th dual space $I^{(n)}$ is zero; i.e., $H^1({\cal A},I^{(n)})=\{0\}$. Further, $\cal A$ is n-ideally amenable if $\cal A$ is $n-I-$weakly amenable for every closed two-sided ideal $I$ in $\cal A$. We find some relationships of $n-I-$ weak amenability and $m-J-$ weak amenability for some different m and n or for different closed ideals $I$ and $J$ of $\c…
This paper is devoted to the study of unbounded derivations on Banach quasi *-algebras with a particular emphasis to the case when they are infinitesimal generators of one parameter automorphisms groups. Both of them, derivations and automorphisms are considered in a weak sense; i.e., with the use of a certain families of bounded sesquilinear forms. Conditions for a weak *-derivation to be the generator of a *-automorphisms group are given.
This note aims to highlight the link between representable functionals and derivations on a Banach quasi *-algebra, i.e. a mathematical structure that can be seen as the completion of a normed *-algebra in the case the multiplication is only separately continuous. Representable functionals and derivations have been investigated in previous papers for their importance concerning the study of the structure properties of a Banach quasi *-algebra and applications to quantum models.
Introducing the notions of (inner) $σ$-derivation, (inner) $σ$-endomorphism and one-parameter group of $σ$-endomorphisms ($σ$-dynamics) on a Banach algebra, we correspond to each $σ$-dynamics a $σ$-derivation named as its $σ$-infinitesimal generator. We show that the $σ$-infinitesimal generator of a $σ$-dynamics of inner $σ$-endomorphisms is an inner $σ$-derivation and deal with the converse. We also establish a nice generalized Leibniz formula and extend the Kleinenckr--Sirokov theorem for $σ$-derivations under certain conditions.
Read produced the first example of a Banach space $E_{\text{R}}$ such that the associated Banach algebra $\mathscr{B}(E_{\text{R}})$ of bounded operators admits a discontinuous derivation (J. London Math. Soc. 1989). We generalise Read's main theorem about $\mathscr{B}(E_{\text{R}})$ from which he deduced this conclusion, as well as the key technical lemmas that his proof relied on, by constructing a strongly split-exact sequence {0} --> $\mathscr{W}(E_{\text{R}})$ --> $\mathscr{B}(E_{\text{R}})$--> $\tilde{\ell_2}$-->{0}, where $\mathscr{W}(E_{\text{R}})$ denotes the ideal of weakly compact o…
Let $n\in \Bbb N-\{1\},$ and let $A$ be a Banach algebra. An additive map $D: A\to A$ is called n-Jordan derivation if $$D(a^n)=D(a)a^{n-1}+aD(a)a^{n-2}+...+a^{n-2}D(a)a+a^{n-1}D(a),$$ for all $a \in {A}$. Using fixed point methods, we investigate the stability of n--Jordan derivations (n--Jordan $^*-$derivations) on Banach algebras ($C^*-$algebras). Also we show that to each approximate $^*-$Jordan derivation $f$ in a $C^*-$ algebra there corresponds a unique $^*-$derivation near to $f$.
Let $B$ be a Banach $A-bimodule$ and let $n\geq 0$. We investigate the relationships between some cohomological groups of $A$, that is, if the topological center of the left module action $π_\ell:A\times B\rightarrow B$ of $A^{(2n)}$ on $B^{(2n)}$ is $B^{(2n)}$ and $H^1(A^{(2n+2)},B^{(2n+2)})=0$, then we have $H^1(A,B^{(2n)})=0$, and we find the relationships between cohomological groups such as $H^1(A,B^{(n+2)})$ and $H^1(A,B^{(n)})$, spacial $H^1(A,B^*)$ and $H^1(A,B^{(2n+1)})$. We obtain some results in Connes-amenability of Banach algebras, and so for every compact group $G$, we conclude t…
We identify concrete examples of hypercyclic generalised derivations acting on separable ideals of operators and establish some necessary conditions for their hypercyclicity. We also consider the dynamics of elementary operators acting on particular Banach algebras, which reveals surprising hypercyclic behaviour on the space of bounded linear operators on the Banach space constructed by Argyros and Haydon.
In this paper, first we give a simple criterion for the Arens regularity of a bilinear mapping on normed spaces, which applies in particular to Banach module actions and then we investigate those conditions under which the second adjoint of a derivation into a dual Banach module is again a derivation. As a consequence of the main result, a simple and direct proof for several older results is also included.
In this paper we define the module extension dual Banach algebras and we use this Banach algebras to finding the relationship between $weak^*-$continuous homomorphisms of dual Banach algebras and Connes-amenability. So we study the $weak^*-$continuous derivations on module extension dual Banach algebras.
A Banach algebra $A$ is said to be zero Lie product determined if every continuous bilinear functional $φ\colon A\times A\to \mathbb{C}$ satisfying $φ(a,b)=0$ whenever $ab=ba$ is of the form $φ(a,b)=ω(ab-ba)$ for some $ω\in A^*$. We prove that $A$ has this property provided that any of the following three conditions holds: (i) $A$ is a weakly amenable Banach algebra with property $\mathbb{B}$ and having a bounded approximate identity, (ii) every continuous cyclic Jordan derivation from $A$ into $A^*$ is an inner derivation, (iii) $A$ is the algebra of all $n\times n$ matrices, where $n\ge 2$, …
In this paper, we prove some theorems related to properties of generalized symmetric hybrid mappings in Banach spaces. Using Banach limits, we prove a fixed point theorem for symmetric generalized hybrid mappings in Banach spaces. Moreover, we prove some weak convergence theorems for such mappings by using Ishikawa iteration method in a uniformly convex Banach space.
In general Banach spaces, the metric projection map lacks the powerful properties it enjoys in Hilbert spaces. There are a few generalized projections that have been proposed in order to resolve many of the deficiencies of the metric projection. However, such notions are predominantly studied in Banach spaces with rich topological structures, such as uniformly convex Banach spaces. In this paper, we investigate two notions of generalized projection in general Banach spaces. Various examples are provided to demonstrate the proposed notions and the loss of structure in the generalized projection…
Let A be a Banach algebra and let X be a Banach A -bimodule. In studying the bounded Hochschild cohomology groups H^1(A,X) it is often useful to extend a given derivation D: A-> X to a Banach algebra B containing A as an ideal, thereby exploiting (or establishing) hereditary properties. This is usually done using (bounded/unbounded) approximate identities to obtain the extension as a limit of operators b->D(ba)-b.D(a), (a in A) in an appropriate operator topology, the main point in the proof being to show that the limit map is in fact a derivation. In this paper we make clear which part of thi…
The objective of this paper is to study the existence of the generalized Drazin inverse of the sum $a+b$ in a Banach algebra and present explicit expressions for the generalized Drazin inverse of this sum, under new conditions.
Let $A$ be a Banach algebra and $M$ be a Banach right $A$-module. A linear map $δ: M\to M$ is called a generalized derivation if there exists a derivation $d : A \to A$ such that $$δ(xa)=δ(x)a + x d(a) \quad (a \in A, x \in M).$$ In this paper, we associate a triangular Banach algebra ${\mathcal T}$ to Banach $A$-module $M$ and investigate the relation between generalized derivations on $M$ and derivations on ${\mathcal T}$. In particular, we prove that the so-called generalized first cohomology group of $M$ is isomorphic to the first cohomology group of ${\mathcal T}$.
In this paper, we introduce the concept of j-hom-derivation, $j\in\{1,2\}$ and solve the new generalized additive-quadratic functional equations in the sense of ternary Banach algebras. Moreover, using the fixed point method, we prove its Hyers-Ulam stability.
Concepts of g-fusion frame and gf-Riesz basis in a Hilbert to a Banach space is being presented. Some properties of g-fusion frame and gf-Riesz basis in Banach space have been developed. We discuss perturbation results of g-fusion frame in a Banach space. Finally, we construct g-p-fusion frames in Cartesian product of Banach spaces and tensor product of Banach spaces.
In this paper, the split common null point problem in two Banach spaces is considered. Then, using the generalized resolvents of maximal monotone operators and the generalized projections and an infinite family of nonexpansive mappings, a strong convergence theorem for finding a solution of the split common null point problem in two Banach spaces in the presence of a sequence of errors will be proved.
We prove several results concerning the representation of projections on arbitrary Banach spaces. We also give illustrative examples including an example of a generalized bi-circular projection which can not be written as the average of the identity with an isometric reflection. We also characterize generalized bi-circular projections on $C_0(\Om,X)$, with $\Om$ a locally compact Hausdorff space (not necessarily connected) and $X$ a Banach space with trivial centralizer.
In this paper, we study a new iterative method for finding the fixed point of a weak Bregman relatively nonexpansive mapping and the set of solutions of generalized mixed equilibrium problems in Banach spaces.
In this paper, two generalized algorithms for solving the variational inequality problem in Banach spaces are proposed. Then the strong convergence of the sequences generated by these algorithms will be proved under the suitable conditions. Finally, using MATLAB software, we provide some numerical examples to illustrate our results.
Let ${\mathcal N}$ be a nest on a complex Banach space $X$ and let $\mbox{ Alg}{\mathcal N}$ be the associated nest algebra. We say that an operator $Z\in \mbox{ Alg}{\mathcal N}$ is an all-derivable point of $\mbox{ Alg}{\mathcal N}$ if every linear map $δ$ from $\mbox{ Alg}{\mathcal N}$ into itself derivable at $Z$ (i.e. $δ$ satisfies $δ(A)B+Aδ(B)=δ(Z)$ for any $A,B \in \mbox{ Alg}{\mathcal N}$ with $AB=Z$) is a derivation. In this paper, it is shown that every injective operator and every operator with dense range in $\mbox{Alg}{\mathcal N}$ are all-derivable points of $\mbox{Alg}{\mathcal …
For each ordinal $α< ω_1$, we prove the existence of a separable, reflexive Banach space with a basis and Szlenk index $ω^{α+1}$ which is universal for the class of separable, reflexive Banach spaces $X$ such that the Szlenk indices $Sz(X), Sz(X^*)$ do not exceed $ω^α$.
In this paper, we investigate a new notion of accretive mappings called generalized $αβ$-$H((.,.),(.,.))$-mixed accretive mappings in Banach spaces. We extend the concept of proximal-point mappings associated with generalized $m$-accretive mappings to the generalized $αβ$-$H((.,.),(.,.))$-mixed accretive mappings and prove that the proximal-point mapping associated with generalized $αβ$-$H((.,.),(.,.))$-mixed accretive mapping is single-valued and Lipschitz continuous. Some examples are given to justify the definition of generalized $αβ$-$H((.,.),(.,.))$-mixed accretive mappings. Further, by u…
In this paper, we present the generalization of B-contraction and C-contraction due to Sehgal and Hicks respectively. We also study some properties of C-contraction in probabilistic metric space.
In this paper, we investigate the various different generalized inverses in a Banach algebra with respect to prescribed two idempotents $p$ and $q$. Some new characterizations and explicit representations for these generalized inverses, such as $a^{(2)}_{p,q}$, $a^{(1,2)}_{p,q}$ and $a^{(2,l)}_{p,q}$ will be presented. The obtained results extend and generalize some well--known results for matrices or operators.
In this paper, we achieve the general solution and the generalized Hyers-Ulam-Rassias stability of the following functional equation $$f(x+ky)+f(x-ky)=k^2f(x+y)+k^2f(x-y)+2(1-k^2)f(x)\eqno {2 cm}$$for fixed integers $k$ with $k\neq0,\pm1$ in the quasi-Banach spaces.
