Classification of group actions and structure of transformation group C*-algebras

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This project has two main lines of investigation. The first is the classification of actions of finite groups on Kirchberg algebras (purely infinite, simple, separable, nuclear C*-algebras). A theorem very similar to the classification of Kirchberg algebras satisfying the universal coefficient theorem should be possible at the very least when the group is cyclic of prime order, the action is in a suitable bootstrap class (as is required for the classification of algebras), and the action is pointwise outer. One expects classification up to conjugacy, not just cocycle conjugacy. Various extensions of this expected result are also under investigation. The second avenue of research is the study of the transformation group C*-algebras of free minimal actions of the group of integers, and the product of several copies of the group of integers, on compact metric spaces. In all cases, the objective is to describe the structure of these algebras. For the integers acting on an infinite dimensional space, the principal investigator seeks to relate the mean dimension of the action to the radius of comparison of the crossed product. For the product of copies of the integers acting smoothly on a compact manifold (and in some other cases), the principal investigator hopes to prove that the transformation group C*-algebra is stable under tensoring with the Jiang-Su algebra and that it has related good behavior. For the same group, but now acting on the Cantor set, the principal investigator would like to prove that the transformation group C*-algebra has tracial rank zero.This project would have a number of broader impacts. First, one of the proposed results would provide a new and striking link with another part of mathematics. Second, previous work of the principal investigator has connections with the study of the quantum mechanical behavior of an electron in a quasicrystal, and some of the work in this project has the potential to shed further light on this problem. Third, the principal investigator is active in supervising graduate students at one of the few institutions in the Pacific Northwest with a substantial Ph.D. program. Moreover, he is serving as an informal coadvisor to students at two other universities. Fourth, through continued research interaction with former graduate students, the principal investigator expects to help support mathematical research at primarily undergraduate institutions and to foster international cooperation with Mexico. Fifth, some elementary questions and numerical experiments related to the project can become undergraduate senior theses at the principal investigator's university. Such projects expose undergraduates to mathematical research, and thus advance the overall technological preparedness of the U.S.