scholarly journals Biderivations and bihomomorphisms in Banach algebras

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2317-2328 ◽  
Author(s):  
Choonkil Park
Keyword(s):  
Complex Number ◽  
Banach Algebras ◽  
Functional Inequalities ◽  
Ulam Stability ◽  
C Algebras ◽  
Nonzero Complex Number

In this paper, we solve the following bi-additive s-functional inequalities || f(x+y,z+w) + f(x+y,z-w)+f(x-y,z+w) + f (x-y,z-w)- 4f(x,z)||? ||s(2f(x+y,z-w)+ 2f(x-y,z + w)- 4f(x,z) + 4f(y,w)||(1) and ||2f(x+y,z-w) + 2f(x-y,z+w)-4f(x,z) + 4f(y,w)|| (2)? ||s(f(x+y,z+w)+ f(x+y,z-w) + f(x-y,z+w)+f(x-y,z-w)-4f(x,z))||, where s is a fixed nonzero complex number with |s| < 1. Moreover, we prove the Hyers-Ulam stability of biderivations and bihomomorphismsions in Banach algebras and unital C+-algebras, associated with the bi-additive s-functional inequalities (1) and (2).

scholarly journals Additive ρ-functional inequalities in β-homogeneous normed spaces

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1651-1658
Author(s):  
Choonkil Park
Keyword(s):  
Banach Spaces ◽  
Complex Number ◽  
Functional Equations ◽  
Additive Functional ◽  
Functional Inequalities ◽  
Ulam Stability ◽  
Normed Spaces ◽  
Homogeneous Complex ◽  
Fixed Complex

In this paper, we solve the following additive ?-functional inequalities ||f (x + y) - f (x) - f (y)|| ? ???(2f (x+y/2) - f(x) + -f (y))??, (1) where ? is a fixed complex number with |?|<1, and ??2f(x+y/2)-f(x)- f(y)???||?(f(x+y)-f(x)-f(y))||, (2) where ? is a fixed complex number with |?|<1/2 , and prove the Hyers-Ulam stability of the additive ?-functional inequalities (1) and (2) in ?-homogeneous complex Banach spaces and prove the Hyers-Ulam stability of additive ?-functional equations associated with the additive ?-functional inequalities (1) and (2) in ?-homogeneous complex Banach spaces.

scholarly journals Hyers–Ulam stability of functional inequalities: a fixed point approach

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Afshan Batool ◽  
Sundas Nawaz ◽  
Ozgur Ege ◽  
Manuel de la Sen
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Banach Algebras ◽  
Fixed Point Method ◽  
Point Method ◽  
Functional Inequalities ◽  
Ulam Stability ◽  
Fixed Point Approach ◽  
Quartic Functional Equation

AbstractUsing the fixed point method, we prove the Hyers–Ulam stability of a cubic and quartic functional equation and of an additive and quartic functional equation in matrix Banach algebras.

COMMENT ON "STABILITY OF (α, β, γ)-DERIVATIONS ON LIE C*-ALGEBRAS" [M. ESHAGHI GORDJI AND N. GHOBADIPOUR, INT. J. GEOM.METHODS MOD. PHYS. 7(7) (2010) 1–10]

2012 ◽  
Vol 10 (02) ◽  
pp. 1220027
Author(s):  
CHOONKIL PARK ◽  
JUNG RYE LEE ◽  
DONG YUN SHIN ◽  
MADJID ESHAGHI GORDJI
Keyword(s):  
Functional Equation ◽  
Ulam Stability ◽  
C Algebras

Eshaghi Gordji and Ghobadipour proved the Hyers–Ulam stability of (α, β, γ)-derivations on Lie C*-algebras associated with the following functional equation [Formula: see text] Under the conditions in the main theorems, we can show that the related mappings must be zero. In this paper, we correct the conditions and prove the corrected theorems.

Bi-additive s-Functional Inequalities and Quasi-∗-Multipliers on Banach ∗-Algebras

2019 ◽  
pp. 199-215
Author(s):  
Jung Rye Lee ◽  
Choonkil Park ◽  
Themistocles M. Rassias
Keyword(s):  
Banach Algebras ◽  
Functional Inequalities

scholarly journals Generalized functional inequalities in Banach spaces

2021 ◽  
Vol 7 (2) ◽  
pp. 337-349
Author(s):  
H. Dimou ◽  
Y. Aribou ◽  
S. Kabbaj
Keyword(s):  
Banach Spaces ◽  
Direct Method ◽  
Additive Functional ◽  
Functional Inequalities ◽  
Ulam Stability

Abstract In this paper, we solve and investigate the generalized additive functional inequalities ‖ F ( ∑ i = 1 n x i ) - ∑ i = 1 n F ( x i ) ‖ ≤ ‖ F ( 1 n ∑ i = 1 n x i ) - 1 n ∑ i = 1 n F ( x i ) ‖ \left\| {F\left( {\sum\limits_{i = 1}^n {{x_i}} } \right) - \sum\limits_{i = 1}^n {F\left( {{x_i}} \right)} } \right\| \le \left\| {F\left( {{1 \over n}\sum\limits_{i = 1}^n {{x_i}} } \right) - {1 \over n}\sum\limits_{i = 1}^n {F\left( {{x_i}} \right)} } \right\| and ‖ F ( 1 n ∑ i = 1 n x i ) - 1 n ∑ i = 1 n F ( x i ) ‖ ≤ ‖ F ( ∑ i = 1 n x i ) - ∑ i = 1 n F ( x i ) ‖ . \left\| {F\left( {{1 \over n}\sum\limits_{i = 1}^n {{x_i}} } \right) - {1 \over n}\sum\limits_{i = 1}^n {F\left( {{x_i}} \right)} } \right\| \le \left\| {F\left( {\sum\limits_{i = 1}^n {{x_i}} } \right) - \sum\limits_{i = 1}^n {F\left( {{x_i}} \right)} } \right\|. Using the direct method, we prove the Hyers-Ulam stability of the functional inequalities (0.1) in Banach spaces and (0.2) in non-Archimedian Banach spaces.

On the Definition of C*-Algebras II

1985 ◽  
Vol 37 (4) ◽  
pp. 664-681 ◽  
Author(s):  
Zoltán Magyar ◽  
Zoltán Sebestyén
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Representation Theorem ◽  
Banach Algebras ◽  
Bounded Operators ◽  
Fundamental Representation ◽  
C Algebras ◽  
Definition Of ◽  
Image Position

The theory of noncommutative involutive Banach algebras (briefly Banach *-algebras) owes its origin to Gelfand and Naimark, who proved in 1943 the fundamental representation theorem that a Banach *-algebra with C*-condition(C*)is *-isomorphic and isometric to a norm-closed self-adjoint subalgebra of all bounded operators on a suitable Hilbert space.At the same time they conjectured that the C*-condition can be replaced by the B*-condition.(B*)In other words any B*-algebra is actually a C*-algebra. This was shown by Glimm and Kadison [5] in 1960.

scholarly journals Differential properties of some dense subalgebras of C*-algebras

1994 ◽  
Vol 37 (3) ◽  
pp. 399-422 ◽  
Author(s):  
E. Kissin ◽  
V. S. Shulman
Keyword(s):  
Banach Algebras ◽  
Sufficient Conditions ◽  
C Algebras ◽  
Differentiable Functions ◽  
Differential Properties

The paper studies some classes of dense *-subalgebras B of C*-algebras A whose properties are close to the properties of the algebras of differentiable functions. In terms of a set of norms on B it defines -subalgebras of A and establishes that they are locally normal Q*-subalgebras. If x = x* ∈ B and f(t) is a function on Sp(x), some sufficient conditions are given for f(x) to belong to B. For p = 1, in particular, it is shown that -subalgebras are closed under C∞-calculus. If δ is a closed derivation of A, the algebras D(δp) are -subalgebras of A. In the case when δ is a generator of a one-parameter semigroup of automorphisms of A, it is proved that, in fact, D(δp) are -subalgebras. The paper also characterizes those Banach *-algebras which are isomorphic to subalgebras of C*-algebras.

scholarly journals ADDITIVE FUNCTIONAL INEQUALITIES AND DERIVATIONS ON HILBERT C*-MODULES

2013 ◽  
Vol 55 (2) ◽  
pp. 341-348 ◽  
Author(s):  
FRIDOUN MORADLOU
Keyword(s):  
Banach Spaces ◽  
Functional Inequality ◽  
Additive Functional ◽  
Functional Inequalities ◽  
Ulam Stability ◽  
Formula Group ◽  
Image Position

AbstractIn this paper we investigate the following functional inequality $ \begin{eqnarray*} \| f(x-y-z) - f(x-y+z) + f(y) +f(z)\| \leq \|f(x+y-z) - f(x)\| \end{eqnarray*}$ in Banach spaces, and employing the above inequality we prove the generalized Hyers–Ulam stability of derivations in Hilbert C*-modules.

scholarly journals Perturbation Theory for Quasinilpotents in Banach Algebras

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1163
Author(s):  
Xin Wang ◽  
Peng Cao
Keyword(s):  
Perturbation Theory ◽  
Banach Algebra ◽  
Perturbation Technique ◽  
Banach Algebras ◽  
C Algebras

In this paper, we prove the following result by perturbation technique. If q is a quasinilpotent element of a Banach algebra and spectrum of p + q for any other quasinilpotent p contains at most n values then q n = 0 . Applications to C* algebras are given.

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