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

Ilwoo Cho () and Palle E. T. Jorgensen ()
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Ilwoo Cho: St. Ambrose University, Department of Mathematics
Palle E. T. Jorgensen: The University of Iowa, Department of Mathematics

Chapter 50 in Operator Theory, 2026, pp 1595-1645 from Springer

Abstract: Abstract 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 . $$\phi .$$

Date: 2026
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DOI: 10.1007/978-3-032-16356-1_45

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