Paterson will investigate the long-standing problem of higher dimensional cohomology for operator algebras, and clarify the difference between amenability and strong amenability for C*- algebras. He will also study amenable C*-algebras in terms of complete boundedness, and investigate the role of amenable foliations in understanding the C*-algebras associated with groups. This work will be based on the recently discovered equivalence of amenability of C*-algebras and the existence of a right invariant mean on a certain space of functions on its unitary group. 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.
