Regularity of Induced Representations and a Theorem of Quigg and Spielberg Astrid an Huef and Iain Raeburn

Mackey’s imprimitivity theorem characterises the unitary representations of a locally compact group G which have been induced from representations of a closed subgroupK; Rieffel’s influential reformulation says that the group C-algebra C(K) is Morita equivalent to the crossed product C0(G/K) × G [14]. There have since been many important generalisations of this theorem, especially by Rieffel [15, 16] and by Green [3, 4]. These are all special cases of the symmetric imprimitivity theorem of [11], which gives a Morita equivalence between two crossed products of induced C-algebras. Quigg and Spielberg proved in [10], by ingenious but indirect methods, that the symmetric imprimitivity theorem, and hence all the other generalisations of Rieffel’s imprimitivity theorem, have analogues for reduced crossed products. A different Morita equivalence between the same reduced crossed products was obtained by Kasparov [7, Theorem 3.15]. Here we identify the representations which induce to regular representations under the Morita equivalence of the symmetric imprimitivity theorem (see Theorem 1 and Corollary 6), and thus obtain a direct proof of the theorem of Quigg and Spielberg (see Corollary 3). We discovered Theorem 1 while trying to understand why Rieffel’s theory of proper actions in [17] gives an equivalence involving reduced crossed products rather than full ones. Theorem 1 has several other interesting applications. We can use it to see, albeit somewhat indirectly, that regular representations themselves nearly always induce to regular representations (see Corollary 7), and it gives a new proof of the main theorem of [5, §4] which avoids a complicated argument involving a composition of crossed-product Morita equivalences (see Corollary 11). It also sheds light on constructions in [8] and [2], which, in various special cases of the symmetric imprimitivity theorem, yield pairs of regular representations which induce to each other (see Remarks 10).