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\(p\)-adic Banach space operators and adelic Banach space operators. (English) Zbl 1428.47032

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.

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
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