Cho, Ilwoo \(p\)-adic Banach space operators and adelic Banach space operators. (English) Zbl 1428.47032 Opusc. Math. 34, No. 1, 29-65 (2014). Summary: In this paper, we study non-Archimedean Banach \(\ast\)-algebras \(\mathfrak M_p\) over the \(p\)-adic number fields \(\mathbb Q_p\), and \(\mathfrak M_{\mathbb Q}\) over the adele ring \(\mathbb A_{\mathbb Q}\). We call elements of \(\mathfrak M_p\), \(p\)-adic operators, for all primes \(p\), respectively, call those of \(\mathfrak M_{\mathbb Q}\), adelic operators. We characterize \(\mathfrak M_{\mathbb Q}\) in terms of \(\mathfrak M_{p}\)’s. Based on such a structure theorem of \(\mathfrak M_{\mathbb Q}\), we introduce some interesting \(p\)-adic operators and adelic operators. Cited in 15 Documents MSC: 47L55 Representations of (nonselfadjoint) operator algebras 05E16 Combinatorial aspects of groups and algebras 11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) 46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis 47L30 Abstract operator algebras on Hilbert spaces Keywords:prime fields; \(p\)-adic number fields; adele ring; \(p\)-adic Banach spaces; adelic Banach space; \(p\)-adic operators; adelic operators PDFBibTeX XMLCite \textit{I. Cho}, Opusc. Math. 34, No. 1, 29--65 (2014; Zbl 1428.47032) Full Text: DOI