Positive-additive functional equations in non-Archimedean $C^*$-algebras
محورهای موضوعی : مجله بین المللی ریاضیات صنعتی
R. Saadati
1
(Department of Mathematics, Iran University of Science and Technology, Tehran, Iran.)
کلید واژه: Functional equation, fixed point, Positive-additive functiona, Linear mapping, Non-Archimedean $C^*$-algebra,
چکیده مقاله :
‎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.
