Simple C*-algebras are basic building blocks in the theory of C*-algebras. The principal investigators, Guihua Gong and Liangqing Li, propose to continue their research on the classification of simple, separable, amenable C*-algebras. They also plan to apply the classification results and the techniques developed in the classification project to study group actions on C*-algebras and differential topology (e.g. the Novikov conjecture on the homotopy invariance of higher signature).
The passage from a finite to an infinite number of degrees of freedom in quantum physics led to the mathematical theory of certain infinite dimensional algebras, called C*-algebras. A C*-algebra is an algebraic system, similar to that of numbers, with its operations of addition, subtraction, multiplication, and division. But unlike the multiplication for numbers, the multiplication in a C*-algebra is not commutative that is, in general, X times Y is not as same as Y times X. This important feature corresponds to Heisenberg uncertainty principle in Quantum Mechanics. The simple C*-algebras are those that cannot be broken into smaller pieces, and in some sense all C*-algebras are built out of them. The investigators propose to work on a complete enumeration (or classification) of simple amenable C*-algebras. They expect that progress on the proposed research will result in important contributions to several mathematical fields including operator algebras, differential topology, and also to the understanding of the infinite dimensional world of quantum physics.
