Korean J. Math. Vol. 22 No. 1 (2014) pp.91-121
DOI: https://doi.org/10.11568/kjm.2014.22.1.91

# Stability of homomorphisms in Banach modules over a $C^*$-Algebra associated with a generalized Jensen type mapping and applications

## Abstract

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

$$2d f(\frac{x_1 + \sum_{j=2}^{2d} (-1)^j x_j}{2d}) - 2d f(\frac{x_1 + \sum_{j=2}^{2d} (-1)^{j+1} x_j}{2d})$$

$$= 2 \sum_{j=2}^{2d} (-1)^j f(x_j) \tag{\ddagger}$$

if and only if the mapping $f : X \rightarrow Y$ is additive, and prove the Cauchy-Rassias stability of the functional equation $\ddagger$ in Banach modules over a unital $C^*$-algebra, and in Poisson Banach modules over a unital Poisson $C^*$-algebra. Let $\mathcal A$ and $\mathcal B$ be unital $C^*$-algebras, Poisson $C^*$-algebras, Poisson $JC^*$-algebras or Lie $JC^*$-algebras. As an application, we show that every almost homomorphism $h : \mathcal A \rightarrow \mathcal B$ of $\mathcal A$ into $\mathcal B$ is a homomorphism when $h(d^n u y) = h(d^n u) h(y)$ or $h(d^n u \circ y) = h(d^n u) \circ h(y)$ for all unitaries $u \in \mathcal A$, all $y \in \mathcal A$, and $n = 0, 1, 2, \cdots$.

Moreover, we prove the Cauchy-Rassias stability of homomorphisms in $C^*$-algebras, Poisson $C^*$-algebras, Poisson $JC^*$-algebras or Lie $JC^*$-algebras, and of Lie $JC^*$-algebra derivations in Lie $JC^*$-algebras.

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