Given a closed subgroup \(H\) in a locally compact group \(G\), representations of \(H\) may be induced to representations of \(G\). This construction simplifies considerably if the subgroup is open. In that case, \(C^*(H)\) is isomorphic to a subalgebra of \(C^*(G)\) and there is a conditional expectation from \(C^*(G)\) onto \(C^*(H)\). This expectation allows to induce representations of \(C^*(H)\) to \(C^*(G)\), and this gives the usual induced representations of Mackey.
J. Kustermans [J. Funct. Anal. 194, No. 2, 410–459 (2002; Zbl 1038.46057)] and S. Vaes [J. Funct. Anal. 229, No. 2, 317–374 (2005; Zbl 1087.22005)] independently defined induction procedures for closed quantum subgroups in locally compact quantum groups. Both constructions are rather subtle. This article shows that they are equivalent and much easier for open quantum subgroups. Namely, for an open quantum subgroup, the situation is exactly as for open subgroups: one \(C^*\)-algebra is contained in the other and there is a conditional expectation onto the subalgebra. And the induction process coming from the conditional expectation is equivalent to the induction processes of Kustermans and Vaes.
It is also shown that a representation of a quantum group \(\mathbb{G}\) is induced from an open quantum subgroup \(\mathbb{H}\) if and only if it is covariant for a representation of \(L^\infty(\mathbb{G}/\mathbb{H})\). This is the analogue of Mackey’s imprimitivity theorem in the case of group representations. And it is shown that the Vaes induction preserves weak containment of representations, even if the quantum subgroup is not open.