IJPAM: Volume 107, No. 3 (2016)

THE HECKE ALGEBRA $H(P_{\Q},P_{\Z})$
ARISING IN NUMBER THEORY

Mamoon A. Ahmed
Princess Sumaya University for Technology
Amman, JORDAN


Abstract. The algebra $H(P_{\Q}^{+}, P_{\Z})$ arises in number theory and has been studied in [#!BC!#] of Bost and Connes. Laca and Raeburn continued the study in [#!LR!#] and gave an improvement of the theorem of Bost and Connes. This leads us to consider a closely related algebra $H(P_{\Q},P_{\Z})$ because of its interesting connections with $C^{*}$-algebras and group algebras.

In this paper we give a detailed proof of Laca and Raeburn's theorem. Then we define a new Hecke pair $(P_{\Q},P_{\Z})$ and show that the Hecke algebra $H(P_{\Q},P_{\Z})$ is a universal $*$-algebra generated by the elements $\{\mu_{n} : n\in \N^{*}\}$, $\{e(r) : r \in \Q/\Z \}$ and a new element $u=\Big[\left(% \begin{array}{cc} 1 & 0 \\ 0 & -1 \\ \end{array}% \right)\Big].$

Received: March 2, 2016

AMS Subject Classification: 20C08, 33D80

Key Words and Phrases: Hecke algebras, Hecke pair $(P_{\Q}^{+}, P_{\Z})$, universal $*$-algebra, Hecke algebra $H(P_{\Q},P_{\Z})$

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DOI: 10.12732/ijpam.v107i3.20 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2016
Volume: 107
Issue: 3
Pages: 723 - 748


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