The Structure of Simple Separable Amenable C*-Algebras

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The study of operator algebras was begun by John von Neumann in the late 1920s with the aim of understanding quantum mechanics. Currently, there are two active areas of investigation of operator algebras---von Neumann algebras and C*-algebras. The project involves investigations into the classification of C*-algebras. Here a classification of a collection of complicated objects provides a simpler way to identify and distinguish between objects in the more complicated class. The classification theory for von Neumann algebras is relatively well developed. The classification was obtained by Connes (and others) in the 1970s. While on the C*-algebra side, it was only in the 1990s that one began to see rapid developments in classification theory. Since then, the classification of C*-algebras has blossomed into one of the central areas in the study of C*-algebras, and tremendous progress has been made in the last few years. The research in this project is to push the classification theory even further, and to seek more applications or interactions of the classification theory with other areas of mathematics, in particular with the study of dynamical systems.Compared to the classification of unital C*-algebras, the classification of non-unital C*-algebras is far from complete. The principal investigator plans to use the strategy and techniques from the classification of unital simple C*-algebras to push forward the classification program for non-unital simple C*-algebras. In particular, this involves showing that an abstract non-unital C*-algebra can be tracially approximated by certain concrete type I C*-algebras (the reduction theorem), developing a classification theory for this semi-concrete class of C*-algebras, and then studying how these might be put to work together to obtain a full classification for abstract non-unital C*-algebras. The other purpose of the proposed research program is to continue the investigation on the interplay between the classification program and study of dynamical systems. In particular, the principal investigator would like to continue the study on whether the zero mean topological dimension of the dynamical system (for an action of the group of integers, or of a more general amenable group) would be equivalent to the regularity property of the crossed product C*-algebras. The principal investigator also would like to investigate possible impacts of the recent classification results on the study of the underlying dynamical systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.