In this paper, we study non-Archimedean Banach \(*\)-algebras \(\frak{M}_{p}\) over the \(p\)-adic number fields \(\mathbb{Q}_{p}\), and \(\frak{M}_{\mathbb{Q}}\) over the adele ring \(\mathbb{A}_{\mathbb{Q}}\). We call elements of \(\frak{M}_{p}\), \(p\)-adic operators, for all primes \(p\), respectively, call those of \(\frak{M}_{\mathbb{Q}}\), adelic operators. We characterize \(\frak{M}_{ \mathbb{Q}}\) in terms of \(\frak{M}_{p}\)'s. Based on such a structure theorem of \(\frak{M}_{\mathbb{Q}}\), we introduce some interesting \(p\)-adic operators and adelic operators.