Full and reduced coactions of locally compact groups on C∗-algebras

We survey the results required to pass between full and reduced coactions of locally compact groups on C∗-algebras, which say, roughly speaking, that one can always do so without changing the crossed-product C∗-algebra. Wherever possible we use definitions and constructions that are well documented and accessible to non-experts, and otherwise we provide full details. We then give a series of applications to illustrate the use of these techniques. We obtain in particular a new version of Mansfield's imprimitivity theorem for full coactions, and prove that it gives a natural isomorphism between crossed-product functors defined on appropriate categories.