We introduce a notion of normality for a nested pair of (ergodic) discrete measured equivalence relations of type II t . Such pairs are characterized by a group Q which serves as a quotient for the pair, or by the ability to synthesize the larger relation from the smaller and an action (modulo inner automorphisms) of Q on it; in the case where Q is amenable, one can work with a genuine action. We classify ergodic subrelations of finite index, and arbitrary normal subrelations, of the unique amenable relation of type Hi. We also give a number of rigidity results; for example, if an equivalence relation is generated by a free 11,-action of a lattice in a higher rank simple connected non-compact Lie group with finite centre, the only normal ergodic subrelations are of finite index, and the only strongly normal, amenable subrelations are finite. IntroductionThe purpose of this paper is to study subrelations of ergodic equivalence relations.Let HQG be countable groups, and suppose G acts as measure-preserving automorphisms on a standard space (X, 38, p); we suppose the action (x, g ) e X x G-» xg € X is free, that fi(X) = 1 and that H acts ergodically, so that if E e 38 with Eh = E for all h e H, then / i ( £ ) e {0,1}. We let Sf = {(x,xg):xeX,geG}=>9t={(x,xh):xeX,heH} be the equivalence relations generated by the actions of G and H. In this context, our basic question is: to what extent does the pair 0t s S" 'remember' the pair H s G ? Although this question is too vague to admit a precise answer, we shall see that there are circumstances in which the pair S c^ has a good memory. In particular, we shall show that if H is normal in G, the pair 3? s y determines G/H, and that if H = T is a lattice in a higher rank (simple connected non-compact) Lie group L, and if H is normal in G with G/H amenable and torsion-free, then if determines the rank of L. t To the memory of HENRY ABEL DYE, teacher, friend and colleague.

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Abstract. The main purpose of this paper is to establish a covariant representation theory for coactions of locally compact groups on C*-algebras (including a notion of exterior equivalence), to show how these results extend the usual notions for actions of groups on C*-algebras, and to apply these ideas to classes of coactions termed pointwise unitary and locally unitary to obtain a complete realization of the isomorphism theory of locally trivial principal G-bundles in this context. We are also able to obtain all (Cartan) principal G-bundles in this context, but their isomorphism theory remains elusive.

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We investigate the structure of the C *-algebras O p constructed by Doplicher and Roberts from the intertwining operators between the tensor powers of a representation p of a compact group. We show that each Doplicher-Roberts algebra is isomorphic to a corner in the Cuntz-Krieger algebra OA of a {0,1}-matrix A -A p associated to p. When the group is finite, we can then use Cuntz's calculation of the if-theory of OA to compute K.(O P ).Doplicher and Roberts have recently developed a duality theory for compact subgroups of SU(n, C) in which the dual object consists of a simple C*-algebra OG and an endomorpbism of OG [3,4]. The construction of OG is based on the concrete representation p of G in SU(n,C) rather than the abstract group G, so we prefer to call it O p ; our work originated in an attempt to find out how the structure of O p depends on the choice of representation. To this end we have computed the .fif-theory of O p for finite G, by embedding it as a corner in a Cuntz-Krieger algebra OA , and using Cuntz's calculation of K*{OA) [1]-One conclusion is that different representations of the same finite group can give algebras which have quite different if-theory, and hence are not even stably isomorphic or Morita equivalent.The algebra O p is constructed from the spaces of intertwining operators between the different tensor powers p n of p, and its structure is determined by the decompositions of p n into irreducibles, and hence by the decompositions of TT ® p for •K (= G. The combinatorics of the situation can be summed up in a bipartite graph with G as vertices, and our main observation is that these combinatorics are similar to those involved in Cuntz and Krieger's construction of a C*-algebra OA from a {0,1}-matrix A. When G is compact, A is infinite, and there are technical problems in transferring this combinatorial similarity to the C *-algebra level; indeed, we need to appeal to both [2] and [3] to do it. For finite groups, we can prove directly that O p is a corner in OA , and the simplicity of O p therefore follows from [2] alone. We shall go as far as we can in full generality, since we are optimistic that one can extend the results of [1] to cover infinite A, and use them to compute K»(O P ) for compact G along similar lines.