Galois theory for multiplier Hopf algebras with integrals (Q1294013)

A Hopf Galois theory is developed for regular multiplier Hopf algebras with integrals. The notion of a coaction of a multiplier Hopf algebra on an algebra is introduced, and a Morita context connecting the smash product and the coinvariants is constructed. As in the case of finite dimensional Hopf algebras studied by \textit{M. Cohen}, \textit{D. Fischman} and \textit{S. Montgomery} [in J. Algebra 133, No. 2, 351-372 (1990; Zbl 0706.16023)] and in the case of co-Frobenius Hopf algebras studied by \textit{M. Beattie}, \textit{S. Dăscălescu} and \textit{Ş. Raianu} [in J. Algebra 198, No. 1, 164-183 (1997; Zbl 0901.16017)], a Galois extension can be characterized by the surjectivity of a certain map in the Morita context. The results are applied to prove duality theorems for co-Frobenius Hopf algebras.