Page 48 - Textos de Matemática Vol. 44

38 A. FOSˇNER AND M. SAL MOSLEHIAN
projection map π2 : X ⊕1 A → A is defined by (x, a) → a. For more information about Banach modules we refer the reader to [9].
Our main result can be regarded as an extension of [4, Theorem 2] and [11, Theorem 2].
Theorem 3.1. Let A be a Banach algebra, X a Banach A-bimodule, and ε1, ε2, ν1, ν2 nonnegative real numbers. Suppose that d, g : A → X are mappings satisfying
and
∥d(a+b)−d(a)−d(b)∥≤ε1, ∥g(a+b)−g(a)−g(b)∥≤ε2, ∥d(ab) − d(a)b − ad(b)∥ ≤ ν1, ∥g(ab) − g(a)b − ad(b)∥ ≤ ν2
for all a, b ∈ A. Then there exists a unique derivation D : A → X such that ∥d(a) − D(a)∥ ≤ ε1
for all a ∈ A and a unique generalized derivation G : A → X with an associate derivation D such that
for all a ∈ A. Furthermore,
and
∥g(a) − G(a)∥ ≤ ε2
b(d(a) − D(a)) = (d(a) − D(a))b = 0
(g(a) − G(a))b = 0
for all a, b ∈ A.
Proof. Let us define mappings φd,φg : A → X ⊕1 A by
φd(a) = (d(a), a)
We have ∥φd(a+b)−φd(a)−φd(b)∥
and φg(a) = (g(a), a),
a ∈ A.
= ∥(d(a+b),a+b)−(d(a),a)−(d(b),b)∥ = ∥(d(a+b)−d(a)−d(b),0)∥
= ∥d(a+b)−d(a)−d(b)∥
≤ ε1