Strong morita equivalence of certain transformation groupC*-algebras

Abstract

In the course of extending certain results of Takesaki [11], Nielsen obtained [6] a generalization of a theorem of Mackey [5] and Blattner [1] concerning intertwining operators for induced representations. Nielsen's generalization concerns intertwining operators for the restrictions of induced representations to a subgroup. Now in [7] it was shown that the theorem of Mackey and Blattner is a large part of the statement that the inducing process establishes a Morita equivalence, that is, an equivalence between the representation theories [8], for certain C*-algebras related to induced representations. In fact, the equivalence is a strong Morita equivalence in the sense that it is implemented by what is called in [7] an "imprimitivity bimodule". All this suggests that perhaps Nielsen's theorem is a reflection of the fact that certain algebras associated with the situation he considers are strongly Morita equivalent. We show here that this is in fact the case. Specifically, let G be a locally compact group, and let H and K be closed subgroups of G. Let K act on the left on G/H and let C*(K, G/H) be the corresponding transformation group C*-algebra [4]. Similarly, let H act on the right on K\G and let C*(H,K\G) be the corresponding C*-algebra. We show that there is a natural imprimitivity bimodule establishing a strong Morita equivalence between these two C*-algebras. tn particular, it follows that these two algebras have equivalent representation theories. We then show that Nielsen's theorem follows from this. Our proofs are essentially algebraic -- requiring little more measure theory than those facts about Haar measure needed to define transformation group algebras and their representations. In particular, our results give a proof of Nielsen's theorem which avoids the measure-theoretic details involving liftings of measures in which his proof becomes embroiled. Some of the formulas developed here can be used to motivate a considerably more elementary proof of Takesaki's results [11] concerning generalized commutation relations (as extended by Nielsen). But the theorems obtained here are not needed for that purpose, and so this matter will be treated elsewhere [9].