Browsing by Subject "semicircular elements"

Now showing 1 - 4 of 4

Access status: Open Access ,
Banach *-algebras generated by semicircular elements induced by certain orthogonal projections
(Wydawnictwa AGH, 2018) Cho, Ilwoo; Jørgensen, Palle E.T.
The main purpose of this paper is to study structure theorems of Banach $∗$-algebras generated by semicircular elements. In particular, we are interested in the cases where given semicircular elements are induced by orthogonal projections in a $C^{*}$-probability space.
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.
Access status: Open Access ,
Semicircular elements induced by p-adic number fields
(Wydawnictwa AGH, 2017) Cho, Ilwoo; Jørgensen, Palle E.T.
In this paper, we study semicircular-like elements, and semicircular elements induced by $p$-adic analysis, for each prime $p$. Starting from a $p$-adic number field $\mathbb{Q}_{p}$, we construct a Banach $∗$-algebra $\mathfrak{LS}_{p}$, for a fixed prime $p$, and show the generating elements $Q_{p,j}$ of $\mathfrak{LS}_{p}$ form weighted-semicircular elements, and the corresponding scalar-multiples $\Theta_{p,j}$ of $Q_{p,j}$ become semicircular elements, for all $j\in\mathbb{Z}$. The main result of this paper is the very construction of suitable linear functionals $\tau_{p,j}^{0}$ on $\mathfrak{LS}_{p}$, making $Q_{p,j}$ be weighted-semicircular, for all $j\in\mathbb{Z}$.
Access status: Open Access ,
Spectral properties of certain operators on the free Hilbert space F[H1,...,HN] and the semicircular law
(Wydawnictwa AGH, 2021) Cho, Ilwoo
In this paper, we fix $N$-many $l^2$-Hilbert spaces $H_k$ whose dimensions are $n_{k} \in \mathbb{N}^{\infty}=\mathbb{N} \cup \{\infty\}$, for $k=1,\ldots,N$, for $N \in \mathbb{N}\setminus\{1\}$. And then, construct a Hilbert space $\mathfrak{F}=\mathfrak{F}[H_{1},\ldots,H_{N}]$ induced by $H_{1},\ldots,H_{N}$, and study certain types of operators on $\mathfrak{F}$. In particular, we are interested in so-called jump-shift operators. The main results (i) characterize the spectral properties of these operators, and (ii) show how such operators affect the semicircular law induced by $\bigcup^N_{k=1} \mathcal{B}_{k}$, where $\mathcal{B}_{k}$ are the orthonormal bases of $H_k$, for $k=1,\ldots,N$.