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 Q , we construct semigroup dynamical systems of A 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).

Certain $*$-homomorphisms acting on unital $C^{*}$-probability spaces and semicircular elements induced by $p$-adic number fields over primes $p$

“…In particular, such a characterization is analyzed by the formula (71), and the structure theorem (77). 10. Shifts on P acting on LS A .…”

“…As applications of [8] and [12], free stochastic calculus for our (weighted-)semicircular law(s) was considered in [11]. And we globalize the (weighted-)semicircularity of [8] and [11] to those induced by Adelic analysis in [10].…”

Banach-Space Operators Acting on Semicircular Elements Induced by p-Adic Number Fields over Primes p

“…The classical independence is replaced by the freeness, by replacing measures on sets to linear functionals on algebras. It has various applications not only in pure mathematics (e.g., [17][18][19]), but also in related topics (e.g., [8][9][10][20][21][22][23]).…”