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 such that x = a , where a and b are integers not divisible by p. Then defines a non-Archimedean norm on Q. The completion of Q with respect to metric, which is denoted by, is called p-adic number field. In fact, p is the set of all formal series x = , here j,p are integers The addition and multiplication between any two elements of p are defined naturally. The norm x is a non-Archimedean norm on p and it makes 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.