C*-algebras from actions of congruence monoids




Bruce, Chris

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We initiate the study of a new class of semigroup C*-algebras arising from number-theoretic considerations; namely, we generalize the construction of Cuntz, Deninger, and Laca by considering the left regular C*-algebras of ax+b-semigroups from actions of congruence monoids on rings of algebraic integers in number fields. Our motivation for considering actions of congruence monoids comes from class field theory and work on Bost–Connes type systems. We give two presentations and a groupoid model for these algebras, and establish a faithfulness criterion for their representations. We then explicitly compute the primitive ideal space, give a semigroup crossed product description of the boundary quotient, and prove that the construction is functorial in the appropriate sense. These C*-algebras carry canonical time evolutions, so that our construction also produces a new class of C*-dynamical systems. We classify the KMS (equilibrium) states for this canonical time evolution, and show that there are several phase transitions whose complexity depends on properties of a generalized ideal class group. We compute the type of all high temperature KMS states, and consider several related C*-dynamical systems.



C*-algebras, Semigroup C*-algebras, Operator algebras, Primitive ideals, KMS states, C*-dynamical system, Number fields, Rings of algebraic integers, Congruence monoids, von Neumann algebras, Noncommutative geometry, Faithful representations, Groupoid C*-algebras, Type III_1 factors