Page 47 - Textos de Matemática Vol. 44

ON APPROXIMATE GENERALIZED DERIVATIONS 37
We have
for all a, b ∈ A. Thus, since F is additive, we get
F (a)h(2nb) = F (a(2nb)) = F ((2na)b) = F (2na)h(b) = 2nF (a)h(b)
and, therefore,
F(a)h(b) = F(a)h(2nb), a,b ∈ A, n ∈ N. 2n
Sending n to infinity, we conclude that
F(ab) = F(a)H(b), a,b ∈ A.
Recall also that F(a)H(b) = F(ab) = F(a)h(b) and, hence, F(a)(h(b) − H(b)) = 0
for all a,b ∈ A. Similarly, (f(b) − F(b))H(a) = 0, a,b ∈ A. The proof is completed.
We use Theorem 2.1 to prove the stability of generalized derivations from a Banach algebra into a Banach bimodule. To do this, we consider module exten- sions as a trick. The idea comes from [11], where authors studied approximate derivations.
3. The main theorem
In the following, A will represent a Banach algebra and X a Banach A- bimodule. It is easy to see that X ⊕1 A is a Banach algebra equipped with the l1 -norm
∥(x,a)∥=∥x∥+∥a∥, x∈X,a∈A, (x,a)(y,b)=(x·b+a·y,ab), x,y∈X,a,b∈A.
Here, the symbol ∥·∥ represent the norm on a normed algebra A and the norm on a normed A-bimodule X, and · denotes the module multiplication on X. The algebra X ⊕1 A is called a module extension Banach algebra. Furthermore, the projection map π1 : X ⊕1 A → X is defined by (x,a) → x. Similarly, the
F(ab) = lim f(2nab) = lim f((2na)b) n→∞ 2n n→∞ 2n
= lim
f(2na)h(b) + f(2nab) − f(2na)h(b) 2n
n→∞
= F (a)h(b)
and the product