In this informal report I will present several principles that can be used to compute the cyclic cohomology of various smooth algebras. These include

1. (a) C∞(X), with X a smooth manifold (but by a different method to that of Alain Connes)

2. (b) C∞(X) ⋊ G, where G is a finite group acting by diffeomorphisms on X

3. (c) C∞(X)G or more generally smooth functions on an orbifold

4. (d) C∞(X), where X is a smooth manifold with boundary or even corners

5. (e) S(G), the convolution algebra of Schwartz functions on a reductive Lie group, including in particular the case G = ℝn.

This last example was in fact the main motivation for this work since at the time it was done (in 1984) I was working on a conjecture of Connes concerning a generalisation of the Connes–Moscovici Index Theorem (see [7]). Here one was interested in computing the pairing between certain specific elements of the cyclic cohomology and the K–theory of S(G). The cyclic cocycles were defined by group cocycles, or equivalently by invariant forms on the homogeneous space G/K via the van Est isomorphism; while the elements of K–theory were represented by abstract indices of twisted Dirac operators, exactly as in [20]. The rough scheme of the computation was to find a concrete ‘heat kernel’ formula for a projection representing the abstract index, substitute it into the cocycle formula and then do a Getzler rescaling to obtain the answer in the limit as Planck's constant approached zero.

A symposium was organised by D.E. Evans at the Mathematics Institute, University of Warwick, between 1st October 1986 and 29th October 1987, with support from the Science and Engineering Research Council, on operator algebras and applications and connections with topology and geometry (K-theory, index theory, foliations, differentiable structures, braids, links) with mathematical physics (statistical mechanics and quantum field theory) and topological dynamics.

As part of that programme, a UK-US Joint Seminar on Operator Algebras was held during 20-25 July 1987 at Warwick, with support from SERC and NSF and organised by D.E. Evans and M. Takesaki. These two volumes contains papers, both research and expository articles, from members of that special week, together with some articles by D.B. Abraham, A.L. Carey, and A. Wassermann on work discussed earlier in the year.

We would like to take this opportunity to thank SERC and NSF for their support, and the participants, speakers and authors for their contributions.

INTRODUCTION

In my opinion, the most important general structure question concerning simple C*-algebras is the extent to which the Murray-von Neumann comparison theory for factors is valid in arbitrary simple C*-algebras. In this article, I will discuss this question, which I call the Fundamental Comparability Question, describe a reasonable framework which has been developed to study the problems involved in the question, give a summary of the rather modest progress made to date including some interesting classes of examples, and give some philosophical arguments why the problems are reasonable, interesting, and worth studying.

To simplify the discussion, in this article we will only consider unital C*-algebras. Most of the theory works equally well in the nonunital case, but there are some annoying technicalities which we wish to avoid here.

Much of what we do in this article will be an elaboration of certain aspects of the theory developed in [Bl 4, §5–6], to which the reader may refer for more information.

Before stating the problem in detail, let us introduce some basic notation and terminology:

Definition 1.1.1. Let p and q be projections in a C*-algebra A. Then p is equivalent to q, written p ∼ q, if there is a partial isometry u ∈ A with u*u=p, uu* = q.

A symposium was organised by D.E. Evans at the Mathematics Institute, University of Warwick, between 1st October 1986 and 29th October 1987, with support from the Science and Engineering Research Council, on operator algebras and applications and connections with topology and geometry (K–theory, index theory, foliations, differentiable structures, braids, links) with mathematical physics (statistical mechanics and quantum field theory) and topological dynamics.

As part of that programme, a UK-US Joint Seminar on Operator Algebras was held during 20–25 July 1987 at Warwick, with support from SERC and NSF and organised by D.E. Evans and M. Takesaki. These two volumes contains papers, both research and expository articles, from members of that special week, together with some articles by D.B. Abraham, A.L. Carey, and A. Wassermann on work discussed earlier in the year.

Introduction

The planar Ising model has become one of the most important statistical mechanical systems for the study of phase transitions and critical phenomena. Although there are many rigorous results, such as correlation inequalities, Peierls argument and the Yang-Lee circle theorem to name but three [reviewed by Griffiths, 1971], which are dimensionindependent in their validity and which lead to results of considerable interest, only the planar model to date benefits from the added insights which stem ultimately from Onsager's tour de force [Onsager, 1944]. It is not the purpose of this article to enter into a general review – for this the reader is referred elsewhere [Gallavotti, 1972] but rather to discuss two more mathematical aspects of the development of Onsager's solution. The first item is the Yang-Baxter system of equations for the planar Ising model in zero field with transfer in the (1,1) direction. This work shows that the Clifford-algebraic structure of the exact solution is a natural consequence of the star-triangle equations. The second item is a Fredholm system which turns out to be of crucial importance in understanding surface and interface problems, as well as the pair correlation function.

We introduce a Galois type invariant for the position of a subalgebra inside an algebra, called a paragroup, which has a group-like structure. Paragroups are the natural quantization of (finite) groups. The quantum groups of Drinfeld and Fadeev, as well as quotients of a group by a non-normal subgroup which appear in gauge theory have a paragroup structure.

In paragroups the underlying set of a group is replaced by a graph, the group elements are substituted by strings on the graph and a geometrical connection stands for the composition law. The harmonic analysis is similar to the computation of the partition function in the Andrews-Baxter-Forrester models in quantum statistical mechanics (e.g. harmonic analysis for the paragroup corresponding to the group ℤ2 is done in the Ising model.) The analogue of the Pontryagin van Kampen duality for abelian groups holds in this context, and we can alternatively use as invariant the coupling system, which is similar to the duality coupling between an abelian group and its dual.

We show that for subfactors of finite Jones index, finite depth and scalar centralizer of the Murray-von Neumann factor R the coupling system (or alternatively the paragroup) is a complete conjugacy invariant. In index less than 4 these conditions are always satisfied and the conjugacy classes of subfactors are rigid: there is one for each Coxeter – Dynkin diagram An and D2n and there are two anticonjugate but nonconjugate for each of the E6 and E8 diagrams.

THE INFINITE DIMENSIONAL COMPLEX ORTHOGONAL AND SPIN GROUPS

The overall reference for this material is a preprint (with title “The infinite complex spin groups”) by John Palmer and myself. Background on the general area is provided by the articles of Araki [1], Ruijsenaars [25,26] (note that Araki refers to the Clifford algebra as a ‘self-dual CAR algebra’) and Palmer [13] although I will try to make this exposition self-contained except for proofs. An interesting reference containing some of the results described here is [34] chapter 12 (which I first read after writing the draft for these notes).

The work discussed here is all inspired by the papers of the Kyoto School [31]. From a mathematical viewpoint the main difficulty with their work is the free use, for infinite dimensional spaces, of results whose proofs are only established in the finite dimensional case. In the latter the methods of proof are purely algebraic and may be found in the papers cited earlier in this paragraph. The generalisation of these finite dimensional results is of interest in its own right, independent of any desire to make rigorous the work of the Kyoto School. In fact it offers the prospect of going beyond their results within the purely infinite dimensional framework.

Introduction. In [8] we gave a proof of Connes' fundamental theorem that injectivity implies hyperfiniteness for finite von Neumann algebras. It consisted in first proving a certain FØlner type condition for injective algebras, and then in combining this FØlner condition with a local Rohlin lemma to get a local approximation property. Using a maximality argument, the latter gives the proof. We show here that by using a slightly stronger local Rohlin lemma, one can derive a similar proof of injectivity implies hyperfiniteness from Connes' FØlner type condition ([l],[2]). Although we will use the same ideas as in [8], by reordering some of the steps in [8] the proof will become here more conceptual and shorter. In fact the proof in [8] can also be shortened using- this reordering of the arguments.

Notation. Throughout this paper, M will denote a finite von Neumann algebra with a fixed normal finite faithful trace τ (τ(1) = 1), ||x||2 = τ(x*x)½, x∈M will be the Hilbert norm given by τ, and L2(M,τ), the completion of M in this norm. We let u(M) denote the unitaries in M.

Injectivity. The algebra M is called injective if there exists an M-invariant state Ψo on B(L2(M,τ)), i.e., Ψ0 satisfies Ψo(χT)=Ψo(Tχ) for all χ∈ M, T∈B(L2(M,τ)). Such a state is called a hypertrace on M and may be regarded as the operator algebra analogue of an invariant mean on an amenable group (see [2]).