Operators induced by prime numbers

Abstract

In this paper, we study operator-theoretic and algebraic structures induced by prime numbers. The Adele ring A Q is an algebraic, topological, and measure-theoretic object induced by the p-adic number fields {Qp} p:prime . By determining a Hilbert space H induced by A Q , we study how prime numbers act on H. In particular, we are interested in the case where they act like shift operators on H. The main purpose of this paper is to understand the properties of such shift operators ap induced by prime numbers p. We call them prime multipliers. We characterize the von Neumann algebra vN () generated by {ap} p:prime . As spectral-theoretic application, we compute the free distributional data of self-adjoint elements ap + a * p , for all primes p.