Simple C*-Algebras

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AbstractLinThe main goal of this project is to classify nuclear simple C*-algebras by their K-theoretical data. The principal investigator introduced a notion of tracial topological rank for C*-algebras. It has been established that this new rank is quite useful in classification of nuclear simple C*-algebras. The investigator proposes to classify unital simple nuclear C*-algebras with tracial topological rank no more than one by their K-theoretical data. These C*-algebras include most known nuclear simple C*-algebras. The investigator also proposes to determine tracial topological ranks for many naturally occurred simple C*-algebras.The simplest examples of noncommutative C*-algebras are collections of matrices. In general, C*-algebras could be thought as collections of infinite matrices. In quantum mechanics physical quantities are represented by operators on a Hilbert space i.e. by infinite matrices. C*-algebras appeared as an outgrowth of quantum physics. But C*-algebras also provide the natural framework for generalizing geometry, topology and measure theory in a fundamental noncommutative way of wide applicability. Simple C*-algebras can be viewed as the key to understanding quantum variables. They are the fundamental building blocks of more general C*-algebras as well as the typical noncommutative C*-algebras. This research project is to study how to describe these C*-algebras via a few data which is topological in nature. Immediate applications can be established in the study of dynamical systems.