Free Probability on Hecke Algebras

Cho, Ilwoo; Gillespie, Timothy L.

doi:10.1007/s11785-014-0403-1

Cited by 15 publications

(35 citation statements)

References 8 publications

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“…In such a case, one can understand arithmetic functions as Krein-space operators under certain Krein-space representations. Also, in [3,4] and [7], we considered free-probabilistic structures on a Hecke algebra H(G p ) for primes p.…”

Section: Previewmentioning

confidence: 99%

Adelic analysis and functional analysis on the finite Adele ring

Cho¹

2018

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Abstract.In this paper, we study operator theory on the * -algebra MP , consisting of all measurable functions on the finite Adele ring A Q , in extended free-probabilistic sense. Even though our * -algebra MP is commutative, our Adelic-analytic data and properties on MP 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 Q , we construct the suitable Hilbert-space representation, and study a C * -algebra MP generated by MP under representation. In particular, we focus on operator-theoretic properties of certain generating operators on MP .

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Section: Previewmentioning

confidence: 99%

Adelic analysis and functional analysis on the finite Adele ring

Cho¹

2018

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“…In modern number theory and its applications, p-adic analysis provides important tools not only for studying mathematical analysis, analytic number theory and non-Archimedian analysis (e.g., [1,3,7,9] and [10]), but also for studying geometry at small distances in mathematical quantum physics (e.g., [14]). So, it is interested in both various mathematical fields and related scientific fields.…”

Section: Motivationmentioning

confidence: 99%

“…In this section we review free-probabilistic structures obtained in [7], and main results of [7] will be introduced for our future work.…”

Section: Normal Hecke Probability Spacesmentioning

confidence: 99%

“…The construction of the linear functional ϕ p on H Yp (originally introduced in [7]) is motivated by the canonical traces on group von Neumann algebras (e.g., [11]), and thepoint-evaluation linear functionals on arithmetic functions in the sense of [4][5][6] and [8]. Clearly, the morphism ϕ p is a well-defined linear functional on H Yp , and hence, the pair pH Yp , ϕ p q forms a free probability space in the sense of Section 2.2.…”

Section: Free-probabilistic Models On H Ypmentioning

confidence: 99%

“…In [7], the author and Gillespie considered certain embedded free-probabilistic subalgebras of Hecke algebras induced by p-adic number fields for primes p. And, in [2], the author extended the free-probabilistic representations of [7] to those fully on the given Hecke algebras, and investigated elements of Hecke algebras as operators realized under the representations. Especially, the spectral theory of such Hilbert-space operators was considered in [2].…”

Section: Introductionmentioning

confidence: 99%

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Free probability on Hecke algebras and certain group C^{*}-algebras induced by Hecke algebras

Cho

2016

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Circular-Like and Circular Elements in Free Product Banach $${*}$$ ∗ -Algebras Induced by p-Adic Number Fields $$\mathbb {Q}_{p}$$ Q p Over Primes p

Cho

2019

Applied Mathematical Analysis: Theory, Methods, and Applications

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In this paper, we study weighted-circular, and circular elements in a certain free product Banach * -probability space (LS, τ 0 ) induced by measurable functions on p-adic number fields Q p , for primes p. To do that, we first constructand-consider weighted-semicircular, and semicircular elements in (LS, τ 0 ). From our (weighted-)semicircular elements, we establish (weighted-)circular elements and study their free distributions by computing joint free moments of them and their adjoints. The circular law is re-characterized by joint free moments of our circular elements and their adjoints. More interestingly, our weighted-circularity dictated by p-adic analysis is fully characterized by weights of weighted-semicircular elements containing number-theoretic data obtained from fixed primes p. Keywords Free probability • p-Adic number fields • Banach * -probability spaces • Weighted-semicircular elements • Semicircular elements • Weighted-circular elements • Circular elements 1991 Mathematics Subject Classification 05E15 • 11G15 • 11R47 • 11R56 • 46L10 • 46L54 • 47L30 • 47L55

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Free Probability on Hecke Algebras

Cited by 15 publications

References 8 publications

Adelic analysis and functional analysis on the finite Adele ring

Adelic analysis and functional analysis on the finite Adele ring

Free probability on Hecke algebras and certain group C^{*}-algebras induced by Hecke algebras

Circular-Like and Circular Elements in Free Product Banach $${*}$$ ∗ -Algebras Induced by p-Adic Number Fields $$\mathbb {Q}_{p}$$ Q p Over Primes p

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