Duality of Restriction and Induction for C-coactions

Consider a coaction δ of a locally compact group G on a C-algebra A, and a closed normal subgroup N of G. We prove, following results of Echterhoff for abelian G, that Mansfield’s imprimitivity between A ×δ| G/N and A ×δ G ×δ̂,r N implements equivalences between Mansfield induction of representations from A×G/N to A×G and restriction of representations fromA×G×rN to A × G, and between restriction of representations from A × G to A × G/N and Green induction of representations from A × G to A × G ×r N . This allows us to deduce properties of Mansfield induction from the known theory of ordinary crossed products.