C*-Actions and Takai Duality
Summary
The study of classical dynamical systems deals with actions of locally compact Hausdorff groups on locally compact Hausdorff spaces. It is long recognized that classical dynamical systems can be studied successfully via C*-algebras. A C*-action is a strongly continuous group homomorphism from a locally compact Hausdorff group to the automorphism group of a C*-algebra. The C*-actions involving the commutative C*-algebras precisely model the classical dynamical systems.
Every C*-action gives rise to a C*-algebra, called the crossed product, which encodes a lot of information about the C*-action. Using crossed products and duality theory for abelian locally compact Hausdorff groups one can develop a certain duality theory for C*-actions. Just as Pontryagin duality describes the second dual of a topological group, Takai duality describes the second dual of a C*-action. The idea is that one recovers a C*-action from its crossed product up to tensoring with another C*-action.