A von neumann algebra over the adele ring and the euler totient function

Abstract

In this chapter, relations between calculus on a von Neumann algebra MQ over the Adele ring AΦ, and free probability on a certain subalgebra Φ of the algebra A; consisting of all arithmetic functions equipped with the functional addition and convolution are studied. By showing that the Adelic calculus over AQ is understood as a free probability on a certain von Neumann algebra MQ, the connections with a system of natural free-probabilisticmodels on the subalgebra Φ in A are considered. In particular, the subalgebra Φ is generated by the Euler totient function φ.