Page 52 - Textos de Matemática Vol. 44

42 A. FOSˇNER AND M. SAL MOSLEHIAN
4. Some remarks
Remark 4.1. Let A be a closed subalgebra of a Banach algebra B. If we consider B as a Banach A-bimodule, we have the next direct consequence of Theorem 3.1 which generalizes Theorem 2 in [4].
Corollary 4.2. Let A be a closed subalgebra of a Banach algebra B and ε1, ε2, ν1, ν2 nonnegative real numbers. Assume that d, g : A → B satisfy
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 → B such that ∥d(a) − D(a)∥ ≤ ε1
for all a ∈ A and a unique generalized derivation G : A → B with an associate derivation D such that
∥g(a) − G(a)∥ ≤ ε2
for all a∈A.
Remark 4.3. We say that an additive mapping f : A → X is C-linear if f(λa) = λf(a) for all a ∈ A and all scalars λ ∈ C. Let us denote the set of allcomplexunitsbyΛ,i.e.,Λ={λ∈C : |λ|=1}.Itturnsoutthatevery additive mapping f : A → X satisfying f(λa) = λf(a) for all a ∈ A and all λ ∈ Λ is C-linear. Using this result, we can easily show by a standard argument that if we replace the first two inequalities in Theorem 3.1 with
∥d(λa+b)−λd(a)−d(b)∥≤ε1, ∥g(λa+b)−λg(a)−g(b)∥≤ε2,
where a, b ∈ A and λ ∈ Λ, then the mappings D and G are C-linear.
Remark 4.4. At the end, let us write two superstability results concerning generalized derivations. First, recall that a Banach A-bimodule X is without orderifx=0istheonlyelementinX suchthatAx=0and,similarly,x=0 is the only element in X such that xA = 0.
Corollary 4.5. Let A be a Banach algebra, X a Banach A-bimodule without order, and ε1, ε2, ν1, ν2 nonnegative real numbers. Assume that d, g : A → X satisfy
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 d is a derivation on A and g is a generalized derivation on A with an associate derivation d.