Induced representations and finite volume homogeneous spaces

We prove that the representation of $\text{C}{}^1\text{(G ×}\text{G}\text{H}\text{)}$ induced from the restriction to H of a unitary representation π of G can be constructed directly from π in the framework of Rieffel's theory of induced representations of C¹-algebras, with the inducing process defined by a generalized conditional expectation. We then show, in the general context of Rieffel's theory, that if the induced representation is CCR, so is the original. In a more special situation, which still generalizes that of a conditional expectation onto a subalgebra, and which includes the operation of inducing from an open subgroup and the above-mentioned process when $\text{G}\text{H}$ is of finite volume, we prove that if the induced representation is type I, so is the original, and obtain a result on intertwining operators. This provides a unified treatment, as well as an extension to the nonseparable case, of certain known results on induction and restriction of representations.