C*-algebras from actions of congruence monoids

Abstract: I will give an overview of recent results for semigroup C*-algebras associated with number fields. These results are already interesting in the case where the field is the rational numbers, and I will focus mostly on this case to make everything more explicit and accessible. C*-algebras of full ax+b-semigroups over rings of algebraic integers were first studied by Cuntz, Deninger, and Laca; their construction has since been generalized by considering actions of congruence monoids. Semigroup C*-algebras obtained this way provide an example class of unital, separable, nuclear, strongly purely infinite C*-algebras which, in many cases, completely characterize the initial number-theoretic data. They also carry canonical time evolutions, and the associated C*-dynamical systems exhibit intriguing phenomena. For instance, the striking similarity between the K-theory formula and the parameterization space for the low temperature KMS states, observed by Cuntz in the case of the full ax+b-semigroup, persists in the more general setting. Part of this work is joint with Xin Li, and part is joint with Marcelo Laca and Takuya Takeishi.

functional analysisgroup theoryoperator algebrasquantum algebra

Audience: researchers in the topic

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