Extension of K.M.S. states with respect to some locally compact groups

Given a state ω on the C∗-system {A, R×G, α} which is K.M.S. with respect to time on the sub-algebra of A of fixed points under the action of some locally compact group G, we study its K.M.S.-extension to the whole algebra. In this work, we study more precisely the cases where G is a class of locally compact abelian groups (such that {A, G, α|G} is almost periodic) or a product of these groups with a compact group.