A generalized Jensen’s mapping and linear mappings between Banach modules

Abstract.

Let X and Y be vector spaces. It is shown that a mapping f : X → Y satisfies the functional equation

$$ {\left( {d + 1} \right)}f{\left( {\frac{{{\sum\nolimits_{j = 1}^{d + 1} {x_{j} } }}} {{d + 1}}} \right)} = {\sum\limits_{j = 1}^{d + 1} {f{\left( {x_{j} } \right)}} } $$

(‡)

if and only if the mapping f : X → Y is additive, and prove the Cauchy–Rassias stability of the functional equation (‡) in Banach modules over a unital C*-algebra. Let ${\cal A}$ and ${\cal B}$ be unital C*-algebras, Poisson C*-algebras, Poisson JC*-algebras or Lie JC*-algebras. As an application, we show that every almost homomorphism h : ${\cal A}$ → ${\cal B}$ of ${\cal A}$ into ${\cal B}$ is a homomorphism when h((d + 2)ⁿuy) = h((d + 2)ⁿu)h(y) or h((d + 2)ⁿu ∘ y) = h((d + 2)ⁿu) ∘ h(y) for all unitaries u ∈ ${\cal A}$, all y ∈ ${\cal A}$, and n = 0, 1, 2, • • • .

Moreover, we prove the Cauchy–Rassias stability of homomorphisms in C*-algebras, Poisson C*-algebras, Poisson JC*-algebras or Lie JC*-algebras.