Semigroup C*-algebras of number-theoretic origin

I will give an introduction to semigroup C*-algebras of ax+b-semigroups over rings of algebraic integers in algebraic number fields, a class of C*-algebras that was introduced by Cuntz, Deninger, and Laca. After explaining the construction, I will briefly discuss the state-of-the-art for this example class: These C*-algebras are unital, separable, nuclear, strongly purely infinite, and have computable primitive ideal spaces. In many cases, e.g., for Galois extensions, they completely characterise the underlying algebraic number field.