Positive-additive functional equations in non-Archimedean $C^*$-algebras
الموضوعات : International Journal of Industrial Mathematics
1 - Department of Mathematics, Iran University of Science and Technology, Tehran, Iran.
الکلمات المفتاحية: Functional equation, fixed point, Positive-additive functiona, Linear mapping, Non-Archimedean $C^*$-algebra,
ملخص المقالة :
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 $n_x \in\mathbb{Z}$ 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 $|a_{k}| \le p-1$ 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.
