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This paper has three main purposes; ®rst, to associate to any separable factor M a canonical commuting exact square of groups called the characteristic square ofM; secondly, to formulate at least in a preliminary way, a methodology for approaching ``non-smooth'' classi®cation problems in analysis; and thirdly to use this approach in conjunction with the characteristic square to complete the cocycle conjugacy classi®cation of actions of discrete amenable groups on injective factors of type III1. This result, and previous works on the cocyle conjugacy classi®cation problem combine to produce a cocycle conjugacy classi®cation theorem which applies simultaneously to all countable amenable groups and all injective separable factors, and in which the characteristic square and our new methodology play a crucial role. The cocycle conjugacy classi®cation problem has by now a long and distinguished history, beginning with Connes' ground-breaking outer conjugacy classi®cation of single automorphisms of the approximately ®nite dimensional, or AFD, factor R0 of type II1, [C3]. In the intervening 20 years, enormous progress has been made. The ®rst step was taken by V. Jones, [J], who identi®ed the true nature of the cocycle conjugacy invariants, and completed the classi®cation of actions of ®nite groups on R0. Ocneanu, [O], re®ned this work and introduced new analytic techniques, enabling him to extend the classi®cation to countable amenable groups acting on the semi-®nite AFD factor R0;1, (and to some actions on AFD factors of type III also). Sutherland and Takesaki, using groupoid and duality techniques originating in [JT], extended the classi®cation results to Invent. math. 132, 331±380 (1998)

We announce in this article that i) to each approximately finite dimensional factor R of any type there corresponds canonically a group cohomological invariant, to be called the intrinsic invariant of R and denoted Θ(R), on which Aut(R) acts canonically; ii) when a group G acts on R via α : G 7→ Aut(R), the pull back of Orb(Θ(R)), the orbit of Θ(R) under Aut(R),by α is a cocycle conjugacy invariant of α; iii) if G is a discrete countable amenable group, then the pair of the module, mod(α), and the above pull back is a complete invariant for the cocycle conjugacy class of α. This result settles the open problem of the general cocycle conjugacy classification of discrete amenable group actions on an AFD factor of type III1, and unifies known results for other types.

We investigate the structure of the C*-algebras 𝒪ρ constructed by Doplicher and Roberts from the intertwining operators between the tensor powers of a representation ρ 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 = Aρ associated to ρ. When the group is finite, we can then use Cuntz's calculation of the K-theory of 𝒪A to compute K*(𝒪ρ).

We study group extensions Δ→Γ→Ω, where Γ acts on a C*-algebraA. Given a twisted covariant representation π,V of the pairA, Δ we construct 3-cocycles on Ω with values in the centre of the group generated byV(Δ). These 3-cocycles are obstructions to the existence of an extension of Ω byV(Δ) which acts onA compatibly with Γ. The main theorems of the paper introduce a subsidiary invariant Λ which classifies actions of Γ onV(Δ) and in terms of which a necessary and sufficient condition for the the cohomology class of the 3-cocycle to be non-trivial may be formulated. Examples are provided which show how non-trivial 3-cocycles may be realised. The framework we choose to exhibit these essentially mathematical results is influenced by anomalous gauge field theories. We show how to interpret our results in that setting in two ways, one motivated by an algebraic approach to constrained dynamics and the other by the descent equation approach to constructing cocycles on gauge groups. In order to make comparisons with the usual approach to cohomology in gauge theory we conclude with a Lie algebra version of the invariant Λ and the 3-cocycle.

The main purpose of this paper is to establish a covariant representation theory for coactions of locally compact groups on C ∗ {C^{\ast }} -algebras (including a notion of exterior equivalence), to show how these results extend the usual notions for actions of groups on C ∗ {C^{\ast }} -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 G -bundles in this context. We are also able to obtain all (Cartan) principal G G -bundles in this context, but their isomorphism theory remains elusive.

We introduce a notion of normality for a nested pair of (ergodic) discrete measured equivalence relations of type II1. Such pairs are characterized by a groupQwhich 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) ofQon it; in the case whereQis 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 II1. We also give a number of rigidity results; for example, if an equivalence relation is generated by a free II1-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.

If K is a countable amenable group acting freely and ergodically on a probability space (Γ, μ), and G is an arbitrary countable amenable group, we construct an injection of the space of unitary representations of G into the space of unitary 1-cocyles for K on (Γ, μ); this injection preserves intertwining operators. We apply this to show that for many of the standard non-type-I amenable groups H, the representation theory of H contains that of every countable amenable group.

The purpose of this paper is to provide a complete set of invariants, up to cocycle conjugacy, of the possible actions, a, of a discrete amenable group G on a semifinite injective von Neumann algebra Jt. In the case where Ji = L°° (X', ft), this amounts to giving invariants, up to conjugacy, for the non-singular actions of G on a Lebesgue space (X,fjt), a problem which is unsolved even for G = Z', in the general case, the action of G on the center of JK itself appears as part of the invariant. If Ji is a factor and G = ZP or Z, the problem was solve by fundamental work of A. Gonnes in [2, 4]; refining Connes' techniques, V. Jones, [9], resolved the case where J£ is a factor and G is finite, and A. Ocneanu, [14], resolved the case where Ji is a factor and G is amenable. Also, in [10], Jones and Takesaki gave a complete set of invariants for the case where Jt is no longer a factor, but G is abelian. The main theorem of the present paper subsumes all these results, and depends crucially on the results of Ocneanu in [14], and the techniques developed by Jones and Takesaki in [10] to handle the non-factor case. For technical simplicity, we treat the case where the restriction of a to the center ^(^) is ergodic (i.e. a is centrally ergodic) ; we let

This paper constructs a standard Borel space, PG, and a map p^PG—>G(p} such that each G(p) is a Polish group, and such that every Polish group is isomorphic to at least one of the groups G(p) ; PG thus serves as a parameter space for all Polish groups. We formulate the notion of a Borel map from a standard B Space to Polish groups, and that of a Borel functor from a standard Borel groupoid to Polish groups; both are defined in terms of the existence of Borel factorizations through PG. We apply these ideas to establish a general "Cohomology Lemma," asserting that cocycles, with values in Borel family of Polish groups, may be cobounded into a given family of dense, normal, Borel subgroups, whenever the underlying groupoid is a hyperfinite equivalence relation. The purpose of this paper is to provide a parametrizatio n, by a standard Borel space PG, for the space of Polish topological groups, i. e. those second countable topological groups whose underlying topology may be defined by a complete metric, and to present applications of this to the notion of "Borel functor" from a standard Borel groupoid to Polish groups. The need for such concepts became apparent during the course of joint work with M. Takesaki on the classification of the possible actions (up to cocycle conjugacy) of a discrete amenable group on a hyperfinite, semifinite injective von Neumann algebra [12], and the paper can be viewed as preparatory to this work. However, the point of view adopted also reveals a definition of A. Connes, [4], of the notion of Borel functor from a standard Borel groupoid to standard measure spaces, as being very natural. A common situation in which the problems considered here arise is the following: if G is a Polish group and X a Polish G-space under