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1
Content available Deformation of semicircular and circular laws via p-adic number fields and sampling of primes
Cho Ilwoo, Jorgensen Palie E. T.
Opuscula Mathematica
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2019
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Vol. 39, no. 6
773--813
EN
In this paper, we study semicircular elements and circular elements in a certain Banach *-probability space [formula] induced by analysis on the p-adic number fields Qp over primes p. In particular, by truncating the set P of all primes for given suitable real numbers t < s in R, two different types of truncated linear functionals [formula], and [formula] re constructed on the Banach *-algebra [formula]. We show how original free distributional data (with respect to r°) are distorted by the truncations on P (with respect to [formula], and [formula]). As application, distorted free distributions of the semicircular law, and those of the circular law are characterized up to truncation.
2
Content available Banach *-algebras generated by semicircular elements induced by certain orthogonal projections
Cho I., Jorgensen P. E. T.
Opuscula Mathematica
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2018
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Vol. 38, no. 4
501--535
EN
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.
3
Content available Semicircular elements induced by p-adic number fields
Cho I., Jorgensen P. E. T.
Opuscula Mathematica
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2017
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Vol. 37, no. 5
665--703
EN
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 Qp, we construct a Banach *-algebra [formula], for a fixed prime p, and show the generating elements Qpj of [formula] form weighted-semicircular elements, and the corresponding scalar-multiples Θpj of Qpj become semicircular elements, for all j ∈ Z. The main result of this paper is the very construction of suitable linear functionals [formula] on [formula], making Qpj be weighted-semicircular, for all j ∈ Z.
4
Content available Free probability on Hecke algebras and certain group C*-algebras induced by Hecke algebras
Cho I.
Opuscula Mathematica
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2016
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Vol. 36, no. 2
153--187
EN
In this paper, by establishing free-probabilistic models on the Hecke algebras [formula] induced by p-adic number fields Qp, 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.
5
Content available Certain group dynamical systems induced by Hecke algebras
Cho I.
Opuscula Mathematica
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2016
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Vol. 36, no. 3
337--373
EN
In this paper, we study dynamical systems induced by a certain group [formula] embedded in the Hecke algebra H(Gp) induced by the generalized linear group Gp = GL2(Qp) over the p-adic number fields Qp for a fixed prime p. We study fundamental properties of such dynamical systems and the corresponding crossed product algebras in terms ol free probability on the Hecke algebra H(Gp).
6
Content available remote Gelfand-Raikov representations of Coxeter groups associated with positive definite norm functions
Bożejko M., Hirai T.
Probability and Mathematical Statistics
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2014
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Vol. 34, Fasc. 1
161--180
EN
The main purpose of the paper is to study the type of Gelfand-Raikov representations of Coxeter groups (W, S) for the special positive definite functions coming from the deformed Poisson (Haagerup) positive definite functions qL(w) for some special length (norm) functions L on Coxeter groups W.
7
Content available remote Schwinger-Dyson equations : classical and quantum
Mingo J. A., Speicher R.
Probability and Mathematical Statistics
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2013
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Vol. 33, Fasc. 2
275--285
EN
In this note we want to have another look on Schwinger-Dyson equations for the eigenvalue distributions and the fluctuations of classical unitarily invariant random matrix models. We are exclusively dealing with one-matrix models, for which the situation is quite well understood. Our point is not to add any new results to this, but to have a more algebraic point of view on these results and to understand from this perspective the universality results for fluctuations of these random matrices. We will also consider corresponding non-commutative or “quantum” random matrix models and contrast the results for fluctuations and Schwinger-Dyson equations in the quantum case with the findings from the classical case.
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