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

Document Type: Special issue: Trends in Operators on Banach Spaces



‎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)‎.