Semigroup crossed products and Hecke algebras arising from number fields

Abstract

Recently Bost and Connes considered a Hecke $C^{\ast }$-algebra arising from the ring inclusion of $\mathbb{Z}$ in $\mathbb{Q}$, and a $C^{\ast }$-dynamical system involving this algebra. Laca and Raeburn realized this algebra as a semigroup crossed product, and studied it using techniques they had previously developed for studying Toeplitz algebras. Here we associate Hecke algebras to general number fields, realize them as semigroup crossed products, and analyze their representations.