p-adic Banach space operators and adelic Banach space operators

Abstract

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.