TY  - JOUR
ID  - Cho2018
LB  - Cho2018
AU  - Cho, Ilwoo
TI  - Adelic analysis and functional analysis on the finite Adele ring
ST  - Adelic analysis and functional analysis on the finite Adele ring
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2018
VL  - 38
IS  - 2
SP  - 139
EP  - 185
DO  - https://doi.org/10.7494/OpMath.2018.38.2.139
UR  - https://doi.org/10.7494/OpMath.2018.38.2.139
AB  - 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 \(M_{\mathcal{P}}\).
KW  - representations
KW  - \(C^{*}\)-algebras
KW  - \(p\)-adic number fields
KW  - the Adele ring
KW  - the finite Adele ring
SN  - 1232-9274
ER  -