@Article{Cho2018,
author = "Cho, Ilwoo",
title = "Adelic analysis and functional analysis on the finite Adele ring",
journal = "Opuscula Math.",
year = "2018",
volume = "38",
number = "2",
pages = "139--185",
doi = "https://doi.org/10.7494/OpMath.2018.38.2.139",
url = "https://doi.org/10.7494/OpMath.2018.38.2.139",
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",
issn = "1232-9274"
}