On K-theoretic invariants of semigroup C*-algebras from actions of congruence monoids

We study semigroup $C^*$-algebras of semigroups associated with number fields and initial data arising naturally from class field theory. These turn out to have an interesting $C^*$-algebraic structure, giving access many new examplesof classifiable exhibi ting phenomena which not appeared before. Moreover, using K-theoretic invariants, we investigate how much information about the number-theoretic is encoded in our $C^*$-algebras.