Positive-additive functional equations in non-Archimedean $C^*$-‎algebras

Hensel [K. Hensel, Deutsch. Math. Verein, {6} (1897), 83-88.] discovered the $p$-adic number as a number theoretical analogue of power series in complex analysis. Fix a prime number $p$. for any nonzero rational number $x$, there exists a unique integer $n_x inmathbb{Z}$ such that $x = frac{a}{b}p^{n_x}$, where $a$ and $b$ are integers not divisible by $p$. Then $|x|_p: = p^{-n_x}$ defines a non-Archimedean norm on $mathbb{Q}$. The completion of $mathbb{Q}$ with respect to metric $d(x, y)=|x - y|_p$, which is denoted by $mathbb{Q}_p$, is called {it $p$-adic number field}. In fact, $mathbb{Q}_p$ is the set of all formal series $x = sum_{kgeq n_x}^{infty}a_{k}p^{k}$, where $|a_{k}| le p-1$ are integers. The addition and multiplication between any two elements of $mathbb{Q}_p$ are defined naturally. The norm $Big|sum_{kgeq n_x}^{infty}a_{k}p^{k}Big|_p = p^{-n_x}$ is a non-Archimedean norm on $mathbb{Q}_p$ and it makes $mathbb{Q}_p$ a locally compact field. In this paper, we consider non-Archimedean $C^*$-algebras and, using the fixed point method, we provide an approximation of the positive-additive functional equations in non-Archimedean $C^*$- algebras.