Chun-Gil Park

ON THE STABILITY OF THE LINEAR MAPPING IN BANACH MODULES

Abstract We prove the generalized Hyers–Ulam–Rassias stability of the linear mapping in Banach modules over a unital Banach algebra.

On an approximate automorphism on a C^*-algebra

It is shown that for an approximate algebra homomorphism f : B → B on a Banach *-algebra B, there exists a unique algebra *-homomorphism H: B → B near the approximate algebra homomorphism. This is applied to show that for an approximate automorphism f: A → A on a unital C*-algebra A, there exists a unique automorphism α: A → A near the approximate automorphism. In fact, we show that the approximate automorphism f: A → A is an automorphism.

HOMOMORPHISMS BETWEEN C*-ALGEBRAS ASSOCIATED WITH THE TRIF FUNCTIONAL EQUATION AND LINEAR DERIVATIONS ON C*-ALGEBRAS

It is shown that every almost linear mapping h : A\longrightarrowB of a unital -algebra A to a unital -algebra B is a homomorphism under some condition on multiplication, and that every almost linear continuous mapping h : A\longrightarrowB of a unital -algebra A of real rank zero to a unital -algebra B is a homomorphism under some condition on multiplication. Furthermore, we are going to prove the generalized Hyers-Ulam-Rassias stability of *-homomorphisms between unital -algebras, and of C-linear *-derivations on unital -algebras./ -algebras.

Modified Trif's functional equations in Banach modules over a C∗-algebra and approximate algebra homomorphisms☆

Abstract We prove the Hyers–Ulam–Rassias stability of modified Trifs functional equations in Banach modules over a unital C ∗ -algebra. It is applied to show the stability of Banach algebra homomorphisms between Banach algebras associated with modified Trifs functional equations in Banach algebras.

On the stability of the quadratic mapping in Banach modules

Abstract We prove the generalized Hyers–Ulam–Rassias stability of the quadratic mapping in Banach modules over a unital Banach algebra.

A POINCARÉ-BIRKHOFF-WITT THEOREM FOR POISSON ENVELOPING ALGEBRAS

Poisson algebras have recently become extremely important in many areas of mathematics and have been studied by many people. In particular, Joseph, Vancliff, Hodges and Levasseur proved that symplectic leaves of certain Poisson varieties correspond bijectively to primitive ideals of the corresponding quantum algebras (see [1, 2, 3, 9]). Hence one can ask if there is a Poisson algebraic concept which corresponds to the primitive ideal. In [7], the first author gave a definition of a symplectic ideal in a Poisson algebra as follows: Let R be a Poisson algebra. A Poisson ideal Q of R is said to be symplectic if there is a maximal ideal M of the commutative algebra R such that Q is the largest Poisson ideal contained in M . Moreover, he proved in [7] that there is a one to one correspondence between the primitive ideals of quantum 2×2 matrices algebra and the symplectic ideals of a Poisson algebra constructed appropriately. Note that a primitive ideal of an algebra is the annihilator of a simple module. As a Poisson version for primitive ideals, this paper concerns the question: Is a symplectic ideal of a Poisson algebra R the annihilator of a simple Poisson Rmodule? Let us describe our approach in more details. Let R be a Poisson algebra over a field k and let U(R) be the Poisson enveloping algebra of R. Since a k-vector space M is a Poisson R-module if and only if M is a left U(R)-module by [8, 6],

Linear Functional Equations in Banach Modules over a C*-Algebra

The paper is a survey on the Hyers–Ulam–Rassias stability of linear functional equations in Banach modules over a C*-algebra. Its contents is divided into the following sections: 1. Introduction; 2. Stability of the Cauchy functional equation in Banach modules; 3. Stability of the Jensen functional equation in Banach modules; 4. Stability of the Trif functional equation in Banach modules; 5. Stability of cyclic functional equations in Banach modules over a C*-algebra; 6. Stability of cyclic functional equations in Banach algebras and approximate algebra homomorphisms; 7. Stability of algebra *-homomorphisms between Banach *-algebras and applications.

ON A GENERALIZED TRIF`S MAPPING IN BANACH MODULES OVER A C*-ALGEBRA

Let X and Y be vector spaces. It is shown that a mapping satisfies the functional equation

LINEAR *-DERIVATIONS ON JB*-ALGEBRAS

(\ddagger)\;+mn_{mn-2}C_{k-1}\;\sum\limits_{i=1}^n\;f(\frac {x_{mi-m+1}+...+x_{mi}} {m}) =k\;{\sum\limits_{1{\leq}i_1 is additive, and we prove the Cauchy-Rassias stability of the functional equation in Banach modules over a unital . Let A and B be unital or Lie . As an application, we show that every almost homomorphism h : of A into B is a homomorphism when for all unitaries , and d = 0,1,2,..., and that every almost linear almost multiplicative mapping is a homomorphism when h(2x)=2h(x) for all . Moreover, we prove the Cauchy-Rassias stability of homomorphisms in or in Lie , and of Lie derivations in Lie .

Multi-quadratic mappings in Banach spaces

Abstract It is shown that for a derivation f ( x 1 ˆ * ˆ x j − 1 ˆ x j ˆ x j + 1 ˆ * ˆ x k ) = ∑ j = 1 k x 1 ˆ * ˆ x j − 1 ˆ x j + 1 ˆ * ˆ x k ˆ f ( x j ) on a JB*-algebra B, there exists a unique ℂ-linear *-derivation D : B → B near the derivation.