ABSTRACT
In this chapter, the authors discuss a natural method of inducing an action of a subgroup to an action of the whole group. When the action is a unitary action, the induced action is a unitary action on a homogeneous Hilbert bundle. When one forms the corresponding unitary action on the direct integral of this Hilbert bundle, one obtains the usual notion of induced unitary representation. The authors present the notion of an induced group action. This process constructs actions of a larger group in terms of the actions of a subgroup. The authors consider Borel actions with no quasi-invariant measure. To consider group actions with quasi-invariant measure, they need a different notion of equivalence. A simplified proof based on the behavior of the action of the unitary group on the dual space was later given by Effros.
