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