Free probability on Hecke algebras. (English) Zbl 1325.11127
Summary: In this paper, we establish a free-probabilistic model on the Hecke algebras induced by \(p\)-adic number fields. It provides a new bridge between operator algebra theory and number theory. Based on number-theoretic and algebraic results from Hecke algebra theory, we consider operator-theoretic structures and properties via our free-probabilistic model.
MSC:
| 46L54 | Free probability and free operator algebras |
| 05E15 | Combinatorial aspects of groups and algebras (MSC2010) |
| 46L10 | General theory of von Neumann algebras |
| 46L40 | Automorphisms of selfadjoint operator algebras |
| 46L53 | Noncommutative probability and statistics |
| 47L55 | Representations of (nonselfadjoint) operator algebras |
| 47S10 | Operator theory over fields other than \(\mathbb{R}\), \(\mathbb{C}\) or the quaternions; non-Archimedean operator theory |
Keywords:
free probability; free moments; free cumulants; Hecke algebras; \(p\)-adic number fields; normal Hecke subalgebras; free probability spacesReferences:
| [1] | Bump, D.: Automorphic forms and representations. In: Cambridge Studies in Adv. Math., vol. 55. Cambridge Univ. Press, Cambridge (1996). ISBN: 0-521-65818-7 |
| [2] | Cho, I.: Operators induced by prime numbers. Methods Appl. Math. 19(4), 313-340 (2013) · Zbl 1332.46064 |
| [3] | Cho, \[I.: p\] p -Adic Banach-space operators and adelic Banach-space operators. Opuscula Math. (2013, to appear) · Zbl 1428.47032 |
| [4] | Cho, I.: Classification on arithmetic functions and corresponding free-moment \[L\] L-functions. Bull. Korean Math. Soc. (2013, to appear) · Zbl 1329.11113 |
| [5] | Cho, I.: Free distributional data of arithmetic functions and corresponding generating functions. Compl. Anal. Oper. Theory (2013). doi:10.1007/s11785-013-0331-9 |
| [6] | Cho, I. Gillespie, T.: Arithmetic Functions and Corresponding Free Probability Determined by Primes. Rocky Mt. J. Math., (2013, submitted) |
| [7] | Cho, I., Jorgensen, P. E. T.: Krein-Space Operators Induced by Dirichlet Characters, Special Issues. Contemp. Math. Am. Math. Soc. (2013, to appear) · Zbl 1322.11065 |
| [8] | Cho, I., Jorgensen, P.E.T.: Representations, Krein-space, of arithmetic functions determined by primes. Alg. Rep. Theory (2013, submitted) · Zbl 1305.05233 |
| [9] | Gillespie, T.: Superposition of zeroes of automorphic \[L\] L-functions and functoriality, Univ. of Iowa, PhD Thesis (2010) · Zbl 1251.11035 |
| [10] | Gillespie, T.: Prime number theorems for Rankin-Selberg \[L\] L-functions over number fields. Sci. China Math. 54(1), 35-46 (2011) · Zbl 1251.11035 · doi:10.1007/s11425-010-4137-x |
| [11] | Radulescu, F.: Random matrices, amalgamated free products and subfactors of the \[C^*C\]∗-algebra of a Free Group of Nonsingular Index. Invent. Math. 115, 347-389 (1994) · Zbl 0861.46038 · doi:10.1007/BF01231764 |
| [12] | Speicher, R.: Combinatorial theory of the free product with amalgamation and operator-valued free probability theory. Amer. Math. Soc. Mem., vol. 132, no. 627 (1998) · Zbl 0935.46056 |
| [13] | Voiculescu, D., Dykemma, K., Nica, A.: Free Random Variables, CRM Monograph Series, vol 1, (1992) · Zbl 0795.46049 |
| [14] | Vladimirov, V.S., Volovich, I. V., Zelenov, E. \[I.: p\] p-Adic analysis and mathematical physics, Ser. Soviet & East European Math., vol 1. World Scientific, Singapore (1994). ISBN: 978-981-02-0880-6 · Zbl 0812.46076 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
