Ruan will continue his investigations in operator spaces. This will include the connection with Banach spaces and the application to C*-algebras and von Neumann algebras, non-self- adjoint operator algebras, and applications of operator spaces to quantum groups. The general area of mathematics of this project has its basis in the theory of algebras of Hilbert space operators. Operators can be thought of as finite or infinite matrices of complex numbers. Special types of operators are often put together in an algebra, naturally called an operator algebra. These abstract objects have a surprising variety of applications. For example, they play a key role in knot theory, which in turn is currently being used to study the structure of DNA, and they are of fundamental importance in noncommutative geometry, which is becoming increasingly important in physics.
