Semicircular-like and semicircular laws on Banach $*$-probability spaces induced by dynamical systems of the finite adele ring $A_{\Bbb{Q}}$

Abstract

Starting from the finite adele ring $A_{\Bbb{Q}},$ we construct semigroup dynamical systems of $A_{\Bbb{Q}},$ acting on certain $C^{*}$-probability spaces. From such dynamical-systematic $C^{*}$-probability spaces, we construct Banach-space operators acting on the $C^{*}$-probability spaces and corresponding Banach $*$-probability spaces. In particular, we are interested in Banach-space operators whose free distributions are the (weighted-)semicircular law(s).