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Adelic analysis and functional analysis on the finite Adele ring

Cho I.

Opuscula Mathematica

2018

Vol. 38, no. 2

139--185

In this paper, we study operator theory on the *-algebra Mp, consisting of all measurable functions on the finite Adele ring Aq, in extended free-probabilistic sense. Even though our *-algebra Mp is commutative, our Adelic-analytic data and properties on Mv 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 Aq, we construct the suitable Hilbert-space representation, and study a C*-algebra M-p generated by M-p under representation. In particular, we focus on operator-theoretic properties of certain generating operators on Mp.