Banach Modules Research Articles

Nearly Quadratic n‐Derivations on Non‐Archimedean Banach Algebras

Let n > 1 be an integer, let A be an algebra, and X be an A‐module. A quadratic function D : A → X is called a quadratic n‐derivation if for all a1,…,an ∈ A. We investigate the Hyers‐Ulam stability of quadratic n‐derivations from non‐Archimedean Banach algebras into non‐Archimedean Banach modules by using the Banach fixed point theorem.

Hyers‐Ulam‐Rassias RNS Approximation of Euler‐Lagrange‐Type Additive Mappings

Recently the generalized Hyers‐Ulam (or Hyers‐Ulam‐Rassias) stability of the following functional equation where r1, …, rm ∈ ℝ, proved in Banach modules over a unital C*‐algebra. It was shown that if , ri, rj ≠ 0 for some 1 ≤ i < j ≤ m and a mapping f : X → Y satisfies the above mentioned functional equation then the mapping f : X → Y is Cauchy additive. In this paper we prove the Hyers‐Ulam‐Rassias stability of the above mentioned functional equation in random normed spaces (briefly RNS).

Strong Arens irregularity of bilinear mappings and reflexivity

We provide a sufficient condition for strong (Arens) irregularity of certain bounded bilinear maps, which applies in particular to the adjoint of Banach module actions. We then apply our result to improve several known results concerning to the relation between Arens regularity of certain Banach module actions and reflexivity.

APPROXIMATION OF CUBIC MAPPINGS WITH n-VARIABLES IN β-NORMED LEFT BANACH MODULE ON BANACH ALGEBRAS

Let M = {1, 2, , n} and let V = {}. Denote M\I by for . The goal of this paper is to investigate the solution and the stability using the alternative fixed point of generalized cubic functional equation in -Banach modules on Banach algebras, where with and .

ON STABILITY PROBLEMS WITH SHADOWING PROPERTY AND ITS APPLICATION

Let n 2 be an even integer. We investigate that if an odd mapping f : X ! Y satises the following equation n ∑ j=1 xj r 1 A + ∑ ik2f 0;1g ∑n k=1 ik= n 2 rf ( n ∑ i=1 ( 1) ik xi r ) then f : X ! Y is additive, where r 2 R: We also prove the stability in normed group by using shadowing property and the Hyers-Ulam stability of the functional equation in Banach spaces and in Banach modules over unital C � -algebras. As an application, we show that every almost linear bijection h : A ! B of unital C � -algebras A and B is a C � -algebra isomorphism when h( 2 s rs uy) = h( 2 s rs u)h(y) for all unitaries u 2 A; all y 2 A; and s = 0;1;2;::::

CONTINUITY OF (α,β)-DERIVATIO OF OPERATOR ALGEBRAS

We investigate the continuity of ()-derivations on B(X) or -algebras. We give some sufficient conditions on which ()-derivations on B(X) are continuous and show that each ()-derivation from a unital -algebra into its a Banach module is continuous when and are continuous at zero. As an application, we also study the ultraweak continuity of ()-derivations on von Neumann algebras.

Fixed Points and Stability for Quadratic Mappings in β–Normed Left Banach Modules on Banach Algebras

The goal of the present paper is to investigate some new stability results by applying the alternative fixed point of generalized quadratic functional equation $$\begin{array}{ll}f\left(\sum\limits_{i=1}^{n}a_ix_i\right)+{\sum\limits_{i=1}^{n-1}}{\sum\limits_{j=i+1}^{n}}\left[f(a_ix_i+a_jx_j)+2f(a_ix_i-a_jx_j)\right]\\ \qquad \quad = (3n-2){\sum\limits_{i=1}^{n}}a^2_{i}f(x_{i})\end{array}$$ in β–Banach modules on Banach algebras, where $${a_{1},\dots,a_{n}\in \mathbb{Z}{\setminus}\{0\}}$$ and some $${\ell\in\{1 , 2 ,\dots, n-1\},}$$ a l ≠ ±1 and a n = 1, where n is a positive integer greater or at least equal to two.

On smooth modules

In this note we introduce the notion of smooth module to extend results from homology theory of Banach algebras to the locally convex category. A complete locally convex module over an m-convex algebra is shown to be smooth if and only if it is topologically isomorphic to a reduced inverse limit of Banach modules over Banach algebras. Stability properties for smoothness are discussed and conditions under which an arbitrary locally convex module is rendered smooth are given.

A note on weighted A_p(G)-modules

Let G be a locally compact abelian group and w be a weight function on G. In this paper, we show that the space A_{p,w}(G) is a Banach module over the Figà-Talamanca Herz algebra A_p(G) and study the multiplier space from A_p(G) to A_{p,w}(G).

Character contractibility of Banach algebras and homological properties of Banach modules

Abstract Let A be a Banach algebra and let φ be a nonzero character on A . We give some necessary and sufficient conditions for the left φ -contractibility of A as well as several hereditary properties. We also study relations between homological properties of some Banach left A -modules, the left φ-contractibility and the right φ-amenability of A . Finally, we characterize the left character contractibility of various Banach algebras related to locally compact groups.

A characterization and a generalization of $W^{*}$-modules

We give a new Banach module characterization of W*-modules, also known as self-dual Hilbert C*-modules over a von Neumann algebra. This leads to a generalization of the notion, and the theory, of W * -modules, to the setting where the operator algebras are σ-weakly closed algebras of operators on a Hilbert space. That is, we find the appropriate weak * topology variant of our earlier notion of rigged modules, and their theory, which in turn generalizes the notions of a C * -module and a Hilbert space, successively. Our w*-rigged modules have canonical 'envelopes' which are W*-modules. Indeed, a w*-rigged module may be defined to be a subspace of a W *-module possessing certain properties.

STABILITY OF A MIXED ADDITIVE AND QUADRATIC FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN BANACH MODULES

In this paper we establish the general solution of mixed additive and quadratic functional equation \begin{equation*} f(x+2y) + f(x-2y) + 8f(y) = 2f(x) + 4f(2y) \end{equation*} and investigate the generalized Hyers-Ulam-Rassias stability of this equation in non-Archimedean Banach modules over a unital Banach algebra.

Overconvergent algebraic automorphic forms

I present a general theory of overconvergent p-adic automorphic forms and eigenvarieties for connected reductive algebraic groups G whose real points are compact modulo centre, extending earlier constructions due to Buzzard, Chenevier and Yamagami for forms of GL(n). This leads to some new phenomena, including the appearance of intermediate spaces of "semi-classical" automorphic forms; this gives a hierarchy of interpolation spaces (eigenvarieties) interpolating classical automorphic forms satisfying different finite slope conditions (corresponding to a choice of parabolic subgroup of G at p). The construction of these spaces relies on methods of locally analytic representation theory, combined with the theory of compact operators on Banach modules.

A fixed point method to the generalized stability of a mixed additive and quadratic functional equation in Banach modules

In this paper we establish the general solution of the functional equation and use a fixed point method to prove its Hyers–Ulam–Rassias stability in Banach modules over a unital Banach algebra.

The Taylor spectrum and transversality for a Heisenberg algebra of operators

A problem on noncommutative holomorphic functional calculus is considered for a Banach module over a finite-dimensional nilpotent Lie algebra. As the main result, the transversality property of algebras of noncommutative holomorphic functions with respect to the Taylor spectrum is established for a family of bounded linear operators generating a Heisenberg algebra.Bibliography: 25 titles.

STABILITY OF A MIXED ADDITIVE AND QUADRATIC FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN BANACH MODULES

In this paper we establish the general solution of mixed additive and quadratic functional equation \begin{equation*}\label{5.1} f(x + 2y)+ f(x -2y)+8f(y)=2f(x)+4f(2y)\end{equation*} and investigate the generalized Hyers-Ulam-Rassias stability of this equation in non-Archimedean Banach modules over a unital Banach algebra.

Stability of a 2-Dimensional Functional Equation in a Class of Vector Variable Functions

We prove the Hyers-Ulam stability of a 2-dimensional quadratic functional equation in a class of vector variable functions in Banach modules over a unital -algebra.

On a generalization of W*-modules

a recent paper of the first author and Kashyap, a new class of Banach modules over dual operator algebras is introduced. These generalize the $W^*$-modules (that is, Hilbert $C^*$-modules over a von Neumann algebra which satisfy an analogue of the Riesz

Cauchy-Rassias Stability of Sesquilinear $n$-Quadratic Mappings in Banach Modules

Cauchy-Rassias Stability of Sesquilinear $n$-Quadratic Mappings in Banach Modules

Stability of a mixed functional equation in several variables on Banach modules

In this paper, we establish the general solution of the functional equation ∑ i = 1 n f ( x i − 1 n ∑ j = 1 n x j ) = ∑ i = 1 n f ( x i ) − n f ( 1 n ∑ i = 1 n x i ) ( n ≥ 2 ) , and use a fixed point method to prove its generalized Hyers–Ulam stability in Banach modules over a unital Banach algebra.