Three Bimodules for Mansfield's Imprimitivity Theorem

TL;DR

This paper demonstrates that three naturally associated imprimitivity bimodules related to a maximal coaction of a discrete group on a C*-algebra are isomorphic, unifying different induction processes in operator algebra theory.

Contribution

It proves the isomorphism of three bimodules associated with coactions and shows their inducing maps on representations are identical, linking Mansfield and Green induction.

Findings

All three bimodules are isomorphic.

Inducing maps on representations are identical.

Results extend to twisted coactions and dual actions.

Abstract

There are at least three imprimitivity bimodules naturally associated to a maximal coaction of a discrete group G on a C*-algebra and a normal subgroup of G: Mansfield's bimodule; the bimodule assembled by Ng from Green's imprimitivity bimodule and Katayama duality; and a bimodule assembled from Green's bimodule and a crossed-product Mansfield bimodule. We show that all three of these are isomorphic, so that the corresponding inducing maps on representations are identical. This can be interpreted as saying that Mansfield and Green induction are inverses of one another ``modulo Katayama duality''. These results pass to twisted coactions; dual results starting with an action are also given.