Browsing by Author "Cho, Ilwoo"

Now showing 1 - 13 of 13

Access status: Open Access ,
Adelic analysis and functional analysis on the finite Adele ring
(Wydawnictwa AGH, 2018) Cho, Ilwoo
In this paper, we study operator theory on the $∗$-algebra $\mathcal{M}_{\mathcal{P}}$, consisting of all measurable functions on the finite Adele ring $A_{\mathbb{Q}}$, in extended free-probabilistic sense. Even though our $∗$-algebra $\mathcal{M}_{\mathcal{P}}$ is commutative, our Adelic-analytic data and properties on $\mathcal{M}_{\mathcal{P}}$ are understood as certain free-probabilistic results under enlarged sense of (noncommutative) free probability theory (well-covering commutative cases). From our free-probabilistic model on $A_{\mathbb{Q}}$, we construct the suitable Hilbert-space representation, and study a $C∗$-algebra $M_{\mathcal{P}}$ generated by $\mathcal{M}_{\mathcal{P}}$ under representation. In particular, we focus on operator-theoretic properties of certain generating operators on $M_{\mathcal{P}}$.
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 ,
Certain group dynamical systems induced by Hecke algebras
(2016) Cho, Ilwoo
In this paper, we study dynamical systems induced by a certain group $\mathfrak{T}_{N}^{K}$ embedded in the Hecke algebra $\mathcal{H}(G_{p})$ induced by the generalized linear group $G_{p} = GL_{2}(\mathbb{Q}_{p})$ over the p-adic number fields $\mathbb{Q}_{p}$ for a fixed prime $p$. We study fundamental properties of such dynamical systems and the corresponding crossed product algebras in terms of free probability on the Hecke algebra $\mathcal{H}(G_{p})$.
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 ,
Free probability induced by electric resistance networks on energy Hilbert spaces
(2011) Cho, Ilwoo; Jørgensen, Palle E.T.
We show that a class of countable weighted graphs arising in the study of electric resistance networks (ERNs) are naturally associated with groupoids. Starting with a fixed ERN, it is known that there is a canonical energy form and a derived energy Hilbert space $H_{\mathcal{E}}$. From $H_{\mathcal{E}}$, one then studies resistance metrics and boundaries of the ERNs. But in earlier research, there does not appear to be a natural algebra of bounded operators acting on $H_{\mathcal{E}}$. With the use of our ERN-groupoid, we show that $H_{\mathcal{E}}$ may be derived as a representation Hilbert space of a universal representation of a groupoid algebra $\mathfrak{A}_G$, and we display other representations. Among our applications, we identify a free structure of $\mathfrak{A}_G$ in terms of the energy form.
Access status: Open Access ,
Free probability on Hecke algebras and certain group C*-algebras induced by Hecke algebras
(2016) Cho, Ilwoo
In this paper, by establishing free-probabilistic models on the Hecke algebras $\mathcal{H}\left(GL_{2}(\mathbb{Q}_{p})\right)$ induced by $p$-adic number fields $\mathbb{Q}_{p}$, we construct free probability spaces for all primes $p$. Hilbert-space representations are induced by such free-probabilistic structures. We study $C^{*}$-algebras induced by certain partial isometries realized under the representations.
Access status: Open Access ,
Multi-variable quaternionic spectral analysis
(Wydawnictwa AGH, 2021) Cho, Ilwoo; Jørgensen, Palle E.T.
In this paper, we consider finite dimensional vector spaces $\mathbb{H}^n$ over the ring $\mathbb{H}$ of all quaternions. In particular, we are interested in certain functions acting on $\mathbb{H}^n$, and corresponding functional equations. Our main results show that (i) all quaternions of $\mathbb{H}$ are classified by the spectra of their realizations under representation, (ii) all vectors of $\mathbb{H}^n$ are classified by a canonical extended setting of (i), and (iii) the usual spectral analysis on the matricial ring $M_n(\mathbb{C})$ of all $(n \times n)$-matrices over the complex numbers C has close connections with certain »non-linear« functional equations on $\mathbb{H}^n$ up to the classification of (ii).
Access status: Open Access ,
Non-factorizable C-valued functions induced by finite connected graphs
(2017) Cho, Ilwoo
In this paper, we study factorizability of $C$-valued formal series at fixed vertices, called the graph zeta functions, induced by the reduced length on the graph groupoids of given finite connected directed graphs. The construction of such functions is motivated by that of Redei zeta functions. In particular, we are interested in (i) »non-factorizability« of such functions, and (ii) certain factorizable functions induced by non-factorizable functions. By constructing factorizable functions from our non-factorizable functions, we study relations between graph zeta functions and well-known number-theoretic objects, the Riemann zeta function and the Euler totient function.
Access status: Open Access ,
On dynamical systems induced by p-adic number fields
(2015) Cho, Ilwoo
In this paper, we construct dynamical systems induced by $p$-adic number fields $\mathbb{Q}_{p}$. We study the corresponding crossed product operator algebras induced by such dynamical systems. In particular, we are interested in structure theorems, and free distributional data of elements in the operator algebras.
Access status: Open Access ,
Operators induced by certain hypercomplex systems
(Wydawnictwa AGH, 2023) Alpay, Daniel; Cho, Ilwoo
In this paper, we consider a family $\{ \mathbb{H}_{t}\}_{t\in\mathbb{R}}$ of rings of hypercomplex numbers, indexed by the real numbers, which contain both the quaternions and the split-quaternions. We consider natural Hilbert-space representations $\{(\mathbb{C}^{2},\pi_{t})\}_{t\in\mathbb{R}}$ of the hypercomplex system $\{ \mathbb{H}_{t}\}_{t\in\mathbb{R}}$, and study the realizations $\pi_{t}(h)$ of hypercomplex numbers $h \in \mathbb{H}_{t}$, as $(2\times 2)$-matrices acting on $\mathbb{C}^{2}$, for an arbitrarily fixed scale $t \in \mathbb{R}$. Algebraic, operator-theoretic, spectral-analytic, and free-probabilistic properties of them are considered.
Access status: Open Access ,
p-adic Banach space operators and adelic Banach space operators
(2014) Cho, Ilwoo
In this paper, we study non-Archimedean Banach $∗$-algebras $\frak{M}_{p}$ over the $p$-adic number fields $\mathbb{Q}_{p}$, and $\frak{M}_{\mathbb{Q}}$ over the adele ring $\mathbb{A}_{\mathbb{Q}}$. We call elements of $\frak{M}_{p}$, $p$-adic operators, for all primes $p$, respectively, call those of $\frak{M}_{\mathbb{Q}}$, adelic operators. We characterize $\frak{M}_{\mathbb{Q}}$ in terms of $\frak{M}_{p}$’s. Based on such a structure theorem of $\frak{M}_{\mathbb{Q}}$, we introduce some interesting $p$-adic operators and adelic operators.
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 $k=1,\ldots,N$.