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

‎Hensel [K‎. ‎Hensel‎, ‎Deutsch‎. ‎Math‎. ‎Verein‎, ‎{۶} (۱۸۹۷), ‎۸۳-۸۸.] 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 \in\mathbb{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_{k\geq n_x}^{\infty}a_{k}p^{k}‎, ‎where ‎|a_{k}| \le p-۱ are integers‎. ‎The addition and multiplication‎ ‎between any two elements of \mathbb{Q}_p are defined naturally. ‎The norm \Big|\sum_{k\geq 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.