Browsing by Subject "circular elements"

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Access status: Open Access ,
Deformation of semicircular and circular laws via p-adic number fields and sampling of primes
(Wydawnictwa AGH, 2019) Cho, Ilwoo; Jørgensen, Palle E.T.
In this paper, we study semicircular elements and circular elements in a certain Banach $∗$-probability space $(\mathfrak{LS},\tau ^{0})$ induced by analysis on the $p$-adic number fields $\mathbb{Q}_{p}$ over primes $p$. In particular, by truncating the set $\mathcal{P}$ of all primes for given suitable real numbers $t\lt s$ in $\mathbb{R}$, two different types of truncated linear functionals $\tau_{t_{1}\lt t_{2}}$, and $\tau_{t_{1}\lt t_{2}}^{+}$ are constructed on the Banach $∗$-algebra $(\mathfrak{LS}$. We show how original free distributional data (with respect to $\tau ^{0}$) are distorted by the truncations on $\mathcal{p}$ (with respect to $\tau_{t\lt s}$, and $\tau_{t\lt s}^{+}$). As application, distorted free distributions of the semicircular law, and those of the circular law are characterized up to truncation.