Abstract
In this chapter, relations between calculus on a von Neumann algebra \(\mathfrak{M}_{\mathbb{Q}}\) over the Adele ring \(\mathbb{A}_{\mathbb{Q}}\), and free probability on a certain subalgebra \(\Phi \) of the algebra \(\mathcal{A},\) consisting of all arithmetic functions equipped with the functional addition and convolution are studied. By showing that the Adelic calculus over \(\mathbb{A}_{\mathbb{Q}}\) is understood as a free probability on a certain von Neumann algebra \(\mathfrak{M}_{\mathbb{Q}}\), the connections with a system of natural free-probabilistic models on the subalgebra \(\Phi \) in \(\mathcal{A}\) are considered. In particular, the subalgebra \(\Phi \) is generated by the Euler totient function ϕ.
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Cho, I., Jorgensenb, P.E.T. (2014). A Von Neumann Algebra Over the Adele Ring and the Euler Totient Function. In: Alpay, D. (eds) Operator Theory. Springer, Basel. https://doi.org/10.1007/978-3-0348-0692-3_45-1
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DOI: https://doi.org/10.1007/978-3-0348-0692-3_45-1
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