Opuscula Math. 38, no. 2 (2018), 139-185

Opuscula Mathematica

Adelic analysis and functional analysis on the finite Adele ring

Ilwoo Cho

Abstract. In this paper, we study operator theory on the \(*\)-algebra \(\mathcal{M}_{\mathcal{P}}\), consisting of all measurable functions on the finite Adele ring \(A_{\mathbb{Q}}\), in extended free-probabilistic sense. Even though our \(*\)-algebra \(\mathcal{M}_{\mathcal{P}}\) is commutative, our Adelic-analytic data and properties on \(\mathcal{M}_{\mathcal{P}}\) are understood as certain free-probabilistic results under enlarged sense of (noncommutative) free probability theory (well-covering commutative cases). From our free-probabilistic model on \(A_{\mathbb{Q}}\), we construct the suitable Hilbert-space representation, and study a \(C^{*}\)-algebra \(M_{\mathcal{P}}\) generated by \(\mathcal{M}_{\mathcal{P}}\) under representation. In particular, we focus on operator-theoretic properties of certain generating operators on \(M_{\mathcal{P}}\).

Keywords: representations, \(C^{*}\)-algebras, \(p\)-adic number fields, the Adele ring, the finite Adele ring.

Mathematics Subject Classification: 05E15, 11G15, 11R47, 11R56, 46L10, 46L54, 47L30, 47L55.

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Cite this article as:
Ilwoo Cho, Adelic analysis and functional analysis on the finite Adele ring, Opuscula Math. 38, no. 2 (2018), 139-185, https://doi.org/10.7494/OpMath.2018.38.2.139

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