Opuscula Math. 38, no. 2 (2018), 139-185
https://doi.org/10.7494/OpMath.2018.38.2.139

Opuscula Mathematica

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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