It is reasonable to expect that there are connections between the topological and K-theoretic methods used to study the group C*-algebras and Roe algebras relevant to higher index theory, and the techniques important in the Elliot classification program. However, almost nothing is known here. We propose to initiate a systematic study of these parallels during our visit to the ESI.
An early observation of Winter and Zacharias is that the nuclear dimension of a (uniform) Roe algebra is bounded above by the asymptotic dimension of the underlying metric space. An immediate goal is to prove the opposite inequality. This is the difficult direction: as a rule, it is much harder to go from C*-algebraic hypotheses to geometric conclusions than to pass in the other direction. Špakula and Willett have made some progress on this problem in low-dimensional cases. As Roe C*-algebras are closely related to group C*-algebras and crossed products (the latter are C¤-algebras encoding dynamical systems), we expect to be able to extend our techniques to these C*-algebras as well.
The proposed research is interesting from several other points of view. First, our work would provide the first (non-trivial) examples of C*-algebras with large, but finite, nuclear dimension. Second we expect that it would lead to an intrinsic C*-algebraic proof of Yu’s theorem on groups with finite asymptotic dimension, increasing our understanding of the relevance of C*-algebraic techniques to Novikovtype conjectures. Third, such a proof of Yu’s theorem would also have the potential to increase the scope of the K-theory Universal Coefficient Theorem (UCT) as achieved by similar methods by Higson and Tu – the UCT is an essential ingredient for the classification program, and in the topological understanding of C*-algebras more generally.