School of Mathematics & Statistics

Noncommutative Topology and Noncommutative Geometry

Operator algebras were developed to axiomatise quantum mechanics and can be thought of as a vast generalisation of matrix algebras. C*-algebras are a class of operator algebras which are particularly nice to work with due to their rigid structure. The Gelfand-Naimark Theorem states that every commutative C*-algebra is isometrically isomorphic to the continuous functions on a locally compact Hausdorff space. Using the well-known duality between a topological space and the algebra of continuous functions on it, the Gelfand-Naimark Theorem implies that C*-algebras are the noncommutative analogue of a topological space. Expanding on this line of thought, Alain Connes developed noncommutative geometry and has shown its significance to many fields of mathematics.  Dynamical systems are particularly well suited to the tools of noncommutative topology/geometry as evidenced by their success in providing dynamical invariants in a noncommutative framework.