Browsing by Subject "representations"

Now showing 1 - 4 of 4

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 ,
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 ,
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 ,
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.