Page 49 - Textos de Matemática Vol. 44

ON APPROXIMATE GENERALIZED DERIVATIONS 39
for all a, b ∈ A. Moreover,
∥φd(ab) − φd(a)φd(b)∥ =
for all a, b ∈ A. Similarly, we can show that
∥φg(a + b) − φg(a) − φg(b)∥ ≤ ε2
∥a−(π2 ◦φd)(a)∥
= 1 ∥na−n(π2 ◦φd)(a)∥ n
= 1 ∥(π2 ◦ φd)(na) − (π2 ◦ φd)(na)∥ n
≤ 1 ∥φd(na) − φd(na)∥ n
≤ 1ε1. n
∥(d(ab), ab) − (d(a), a)(d(b), b)∥
= ∥(d(ab) − d(a)b − ad(b), 0)∥
= ∥d(ab) − d(a)b − ad(b)∥ ≤ ν1
and
for all a, b ∈ A. By Theorem 2.1, there exist unique additive mappings φd , φg :
A → X ⊕1 A such that
(3.1) ∥φd(a) − φd(a)∥ ≤ ε1, ∥φg(a) − φg(a)∥ ≤ ε2
for all a ∈ A and
(3.2) φd(ab) = φd(a)φd(b), φg(ab) = φg(a)φd(b)
for all a, b ∈ A. LetusdefinemappingsD,G:A→X by
D(a)=(π1 ◦φd)(a) and G(a)=(π1 ◦φg)(a), a∈A.
We will show that D and G are the appropriate mappings from Theorem 3.1.
We divide the proof into several steps.
Step 1. D is a derivation on A.
Obviously, D is an additive mapping. Now, let n be a positive integer and a ∈ A. Recall that (π2 ◦ φd)(na) = na and (π2 ◦ φd)(na) = n(π2 ◦ φd)(a). Thus, using (3.1), we have
∥φg(ab) − φg(a)φd(b)∥ ≤ ν2