Opuscula Math. 38, no. 2 (2018), 139-185
https://doi.org/10.7494/OpMath.2018.38.2.139
Opuscula Mathematica
Adelic analysis and functional analysis on the finite Adele ring
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.
- I. Cho, Free distributional data of arithmetic functions and corresponding generating functions, Compl. Anal. Oper. Theo. 8 (2014) 2, 537-570.
- I. Cho, Dynamical systems on arithmetic functions determined by prims, Banach J. Math. Anal. 9 (2015) 1, 173-215.
- I. Cho, On dynamical systems induced by \(p\)-adic number fields, Opuscula Math. 35 (2015) 4, 445-484.
- I. Cho, Representations and corresponding operators induced by Hecke algebras, Complex Anal. Oper. Theory 10 (2016) 3, 437-477.
- I. Cho, Free semicircular families in free product Banach \(*\)-algebras induced by \(p\)-adic number fields, Complex Anal. Oper. Theory 11 (2017) 3, 507-565.
- I. Cho, \(p\)-adic number fields acting on \(W^*\)-probability spaces, Turkish J. Anal. Numb. Theo. (2017), to appear.
- I. Cho, T. Gillespie, Free probability on the Hecke algebra, Complex Anal. Oper. Theory 9 (2015) 7, 1491-1531.
- I. Cho, P.E.T. Jorgensen, Semicircular elements induced by \(p\)-adic number fields, Opuscula Math. 37 (2017) 5, 665-703.
- T. Gillespie, Superposition of zeroes of automorphic \(L\)-functions and functoriality, PhD Thesis, Univ. of Iowa, (2010).
- T. Gillespie, Prime Number Theorems for Rankin-Selberg \(L\)-Functions over Number Fields, Sci. China Math. 54 (2011) 1, 35-46.
- F. Radulescu, Random matrices, amalgamated free products and subfactors of the \(C^*\)-algebra of a free group of nonsingular index, Invent. Math. 115 (1994), 347-389.
- R. Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Amer. Math. Soc. Mem. 627 (1998).
- V.S. Vladimirov, I.V. Volovich, E.I. Zelenov, \(p\)-Adic Analysis and Mathematical Physics, Ser. Soviet & East European Math., vol. 1, World Scientific, 1994.
- D. Voiculescu, K. Dykemma, A. Nica, Free Random Variables, CRM Monograph Series, vol. 1, Amer. Math. Soc., Providence, 1992.
- Ilwoo Cho
- Saint Ambrose University, Department of Mathematics and Statistics, 421 Ambrose Hall, 518 W. Locust St., Davenport, Iowa, 52803, USA
- Communicated by P.A. Cojuhari.
- Received: 2017-02-09.
- Revised: 2017-09-12.
- Accepted: 2017-09-26.
- Published online: 2017-12-29.
