Regularity of induced representations and a theorem of Quigg and Spielberg

Abstract

The Symmetric Imprimitivity Theorem provides a Morita equivalence between two crossed products of induced C*-algebras. Quigg and Spielberg proved, by indirect but ingenious methods, that the symmetric imprimitivity theorem has an analog for reduced crossed products. Here we identify the representations which induce to regular representations under the equivalence of the symmetric theorem. We use this result to show that regular representations themselves almost always induce to regular representations. In addition, we obtain a direct proof of the theorem of Quigg and Spielberg.