This project is mathematical research in the theory of operator algebras bearing on issues in geometry and topology. The operator algebras (or C*-algebras) here are thought of as noncommutative analogs of (the algebra of continuous functions on) a compact topological space. Differentiable structure can be introduced in the form of a C*-dynamical system, with a Lie group acting on the C*-algebra. More specifically, one of the goals of the project is to establish a Chern-Weil theory for C*-dynamical systems, thereby introducing a whole new class of differential and topological invariants for noncommutative differentiable manifolds having principal fibre structures. Another problem is to characterize the smooth structures of manifolds (commutative or otherwise) in terms of certain subalgebras and cyclic homology theory. The principal investigator will continue his study of the close relationship between dynamical invariants of structurally stable flows and K-theoretical invariants of flow algebras.
