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

Abstract

‎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.