THE HECKE ALGEBRA \(H(P_{\mathbb{Q}}, P_{\mathbb{Z}})\) AND ITS RELATION TO THE CROSSED PRODUCT \(H(P_{\mathbb{Q}}^{+}, P_{\mathbb{Z}})\) \(\times_{β}\) {1, −1}

Abstract

The algebra  \(H(P_{\mathbb{Q}}^{+}, P_{\mathbb{Z}})\) arose in number theory has been studied by Bost and Connes in [2].  In [1] a related Hecke algebra \(H(P_{\mathbb{Q}}, P_{\mathbb{Z}})\) is considered wherein it is shown to be a universal *-algebra generated by the elements {µ_n: n ∈ N^∗}, {e(r) : r ∈ \(\mathbb{Q}/\mathbb{Z}\)} and an element u = \(\left[ \begin{pmatrix} 1 & 0 \\ 0 & {-1} \\ \end{pmatrix} \right]\).  The goal of this paper is to study the relationship between the Hecke algebra of Bost and Connes and the Hecke algebra \(H(P_{\mathbb{Q}}, P_{\mathbb{Z}})\). By showing the existence of a ∗-automorphism α of \(H(P_{\mathbb{Q}}^{+}, P_{\mathbb{Z}})\), we construct a covariant representation (ι, U) of \(H(P_{\mathbb{Q}}^{+}, P_{\mathbb{Z}})\) \(\times_{β}\) {1, −1} on \(H(P_{\mathbb{Q}}^{+}, P_{\mathbb{Z}})\).  This leads to our main result that \(H(P_{\mathbb{Q}}, P_{\mathbb{Z}})\) is realized as the crossed product \(H(P_{\mathbb{Q}}^{+}, P_{\mathbb{Z}})\) \(\times_{β}\) {1, −1}.