A Von Neumann Algebra over the Adele Ring and the Euler Totient Function

In this chapter, relations between calculus on a von Neumann algebra 𝔐 ℚ $$\mathfrak{M}_{\mathbb{Q}}$$ over the Adele ring 𝔸 ℚ $$\mathbb{A}_{\mathbb{Q}}$$ , and free probability on a certain subalgebra Φ $$\Phi $$ of the algebra 𝒜 , $$\mathcal{A},$$ consisting of all arithmetic functions equipped with the functional addition and convolution are studied. By showing that the Adelic calculus over 𝔸 ℚ $$\mathbb{A}_{\mathbb{Q}}$$ is understood as a free probability on a certain von Neumann algebra 𝔐 ℚ $$\mathfrak{M}_{\mathbb{Q}}$$ , the connections with a system of natural free-probabilistic models on the subalgebra Φ $$\Phi $$ in 𝒜 $$\mathcal{A}$$ are considered. In particular, the subalgebra Φ $$\Phi $$ is generated by the Euler totient function ϕ.