IJPAM: Volume 107, No. 3 (2016)


Mamoon A. Ahmed
Princess Sumaya University for Technology

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(%
1 & 0 \\
0 & -1 \\

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 $H(P_{\Q},P_{\Z})$

Download paper from here.

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

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

$H(P_{\Q},P_{\Z})$ ARISING IN NUMBER THEORY%22&as_occt=any&as_epq=&as_oq=&as_eq=&as_publication=&as_ylo=&as_yhi=&as_sdtAAP=1&as_sdtp=1" title="Click to search Google Scholar for this entry" rel="nofollow">Google Scholar; DOI (International DOI Foundation); WorldCAT.

CC BY This work is licensed under the Creative Commons Attribution International License (CC BY).