Hopf algebras

For every locally compact abelian group G, the set of characters is also a locally compact abelian group and the Pontrjagin theorem says that the double dual is isomorphic to the original group G. For a non-abelian group, a generalized dual can no longer be defined in the form of a group. One has to look for a larger category, namely quantum groups in the form of Hopf algebras. The objective is to study Hopf algebras in the setting of C*-algebras and von Neumann algebras such that a generalization of Pontrjagin duality is obtained.