Page 44 - Textos de Matemática Vol. 44

34 A. FOSˇNER AND M. SAL MOSLEHIAN
number of mathematicians (see [1, 4, 11, 19, 20, 21, 10] and references therein). These results motivated us to investigate approximate generalized derivations. In particular, we reduce the problem of stability of generalized derivations to the problem of stability of ring homomorphisms using an idea of [11] based on the notion of module extension. At the end, additional remarks and results are presented.
2. Preliminary results
In this section we gather some results and notation needed in the sequel. First, by a ring homomorphism we mean an additive mapping on a ring which preserves multiplication. Let A and B be two rings. An additive mapping d : A → B is called a derivation if
d(ab) = d(a)b + ad(b), a, b ∈ A.
If d is a derivation and g : A → B an additive mapping satisfying
g(ab) = g(a)b + ad(b), a, b ∈ A,
then g is called a generalized derivation. This concept was first introduced by Breˇsar in [7] and it is one of the natural generalizations of ordinary derivations. It is easy to see that generalized derivations are exactly those additive mappings g which can be written in the form g = d + f, where d : A → B is a derivation and f : A → B is a left centralizer, i.e., additive mapping with the property f (ab) = f (a)b, a, b ∈ A. For results concerning generalized derivations we refer the reader to [12] and [26].
Let A be a ring, B a Banach algebra, and ε,ν nonnegative real numbers. In 2002, R. Badora [3, Theorem1] proved that if h : A → B is a mapping satisfying ∥h(a + b) − h(a) − h(b)∥ ≤ ε and ∥h(ab) − h(a)h(b)∥ ≤ ν for all a, b ∈ A, then there exists a unique ring homomorphism H : A → B such that ∥h(a) − H(a)∥ ≤ ε for all a ∈ A. We improve this result in the following way.
Theorem 2.1. Let A be a ring, B a Banach algebra, and ε1 , ε2 , ν1 , ν2 nonneg- ative real numbers. Suppose that h, f : A → B are mappings satisfying
∥h(a+b)−h(a)−h(b)∥≤ε1, ∥f(a+b)−f(a)−f(b)∥≤ε2, and
∥h(ab) − h(a)h(b)∥ ≤ ν1, ∥f(ab) − f(a)h(b)∥ ≤ ν2
for all a, b ∈ A. Then there exist unique additive mappings H, F : A → B such
that
for all a∈A and
∥h(a) − H(a)∥ ≤ ε1, ∥f(a) − F(a)∥ ≤ ε2 H(ab) = H(a)H(b), F (ab) = F (a)H(b)