## Final Report Summary - TODYRIC (Topological Dynamics of Rings and C*-algebras)

This project is, within mathematics, of an interdisciplinary nature in that it combines structures and methods from fields as different as functional analysis, topology or number theory. It was concerned with problems in several areas. A starting point was the new concept of a ring C*-algebra associated with a countable ring without zero divisors. For special rings this C*-algebra has a very rich and surprising structure. A particularly interesting case is the ring of algebraic integers in a global field. In this context the C*-algebra contains well known topological dynamical systems. The analysis of ring algebras was used as an organizing principle for the study of many questions in C*-algebra theory, K-theory, ergodic theory and number theory. Some of these questions are well known and very difficult.

The article by Cuntz-Deninger-Laca obtained early on during the funding period not only had a great impact on the further development of the project, but it also initiated a new area of research that found widespread attention by other experts. It relates a ring C*-algebra to the C*-algebra generated by the left regular representation of a semigroup - in this case the so called 'ax+b'-semigroup of the ring. This C*-algebra is functorially associated with the ring. A further very important feature of the algebra is a natural one-parameter automorphism group which carries an intriguing structure of equilibrium (KMS-) states. The KMS- and ground states are determined completely in the paper by Cuntz-Deninger-Laca. The technical difficulties in this study are due to the presence of the class group in the case where the ring is not a principal ideal domain. In that case, for a fixed large inverse temperature, the simplex of KMS-states splits over the class group. The ``partition

functions'' are partial Dedekind zeta-functions for the quotient field of the ring.

The structure of this particular semigroup algebra has also motivated and guided the team member Xin Li in his systematic study of full and regular C*-algebras for semigroups. The point is that semigroup C*-algebras had been studied before, notably by Nica, but for classes of semigroups which, with hindsight, were not interesting and general enough. The study of the more general semigroup C*-algebras, originating from the motivating example of the 'ax+b'-semigroup has led to many new structural insights for these algebras, but also for more general similar algebras. Important new concepts discovered in the course of these investigations are the ones of independence and the Toeplitz condition.

A particularly important new circle of ideas and techniques arose from the study of the K-theory of these algebras. It led to a powerful new method for computing the K-groups, not only for regular C*-algebras of rather general semigroups, but also for quite a large class of similar algebras. Here also the independence and Toeplitz condition play a crucial role. In the special case of the regular C*-algebra for the 'ax+b'-group of the ring of algebraic integers in a number field K, one gets the interesting result that the K-theory is described by a formula that involves the basic number theoretic structure of the number field K, namely the ideal class group and the action of the group of units on the additive group of an ideal.

We obtained essentially complete results on the problems outlined in the grant proposal. In addition our research has opened an entire new area of research and has led to results that were not expected at the time of the proposal.

The article by Cuntz-Deninger-Laca obtained early on during the funding period not only had a great impact on the further development of the project, but it also initiated a new area of research that found widespread attention by other experts. It relates a ring C*-algebra to the C*-algebra generated by the left regular representation of a semigroup - in this case the so called 'ax+b'-semigroup of the ring. This C*-algebra is functorially associated with the ring. A further very important feature of the algebra is a natural one-parameter automorphism group which carries an intriguing structure of equilibrium (KMS-) states. The KMS- and ground states are determined completely in the paper by Cuntz-Deninger-Laca. The technical difficulties in this study are due to the presence of the class group in the case where the ring is not a principal ideal domain. In that case, for a fixed large inverse temperature, the simplex of KMS-states splits over the class group. The ``partition

functions'' are partial Dedekind zeta-functions for the quotient field of the ring.

The structure of this particular semigroup algebra has also motivated and guided the team member Xin Li in his systematic study of full and regular C*-algebras for semigroups. The point is that semigroup C*-algebras had been studied before, notably by Nica, but for classes of semigroups which, with hindsight, were not interesting and general enough. The study of the more general semigroup C*-algebras, originating from the motivating example of the 'ax+b'-semigroup has led to many new structural insights for these algebras, but also for more general similar algebras. Important new concepts discovered in the course of these investigations are the ones of independence and the Toeplitz condition.

A particularly important new circle of ideas and techniques arose from the study of the K-theory of these algebras. It led to a powerful new method for computing the K-groups, not only for regular C*-algebras of rather general semigroups, but also for quite a large class of similar algebras. Here also the independence and Toeplitz condition play a crucial role. In the special case of the regular C*-algebra for the 'ax+b'-group of the ring of algebraic integers in a number field K, one gets the interesting result that the K-theory is described by a formula that involves the basic number theoretic structure of the number field K, namely the ideal class group and the action of the group of units on the additive group of an ideal.

We obtained essentially complete results on the problems outlined in the grant proposal. In addition our research has opened an entire new area of research and has led to results that were not expected at the time of the proposal.