Additive functional inequality in C*-algebras

Resumo

In this paper, we introduce an additive functional inequality
\begin{eqnarray}\label{1}
\left\| 2 g\left(\frac{\lambda u+ y}{2}\right) - \lambda g(u) - g(y) \right\|\le
\left\| s( g\left(\lambda u+ y\right)- \lambda g(u) - g(y) ) \right\|
\end{eqnarray}
for all $\lambda\in \mathbb{C}$, all unitary elements $u$ in a unital $C^*$-algebra $P$ and all $y\in P$, where $|s|<1$.
Using both the direct method and the fixed point method, we establish the Hyers-Ulam stability of inequality (\ref{1}) in unital $C^*$-algebras.
Furthermore, we apply these results to the study of $C^*$-algebra homomorphisms and $C^*$-algebra derivations in unital $C^*$-algebras.