Operator Algebras and Applications

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The objective of the proposed research is to confirm a striking phenomenon. I have discovered that a large class of quite complicated mathematical objects (virtually all naturally arising norm-closed algebras of operators in Hilbert space, including those of interest in mathematical physics, and in other branches of mathematics in which complex systems are considered, such as the theory of dynamical systems and the theory of foliations) can apparently be described in terms of very simple data---often now called the Elliott invariant. This was unexpected, but the evidence is overwhelming. The discovery has far-reaching implications, both for the class of objects involved (the class of amenable C*-algebras), and, it appears likely, for the related areas of mathematics and physics in which these objects appear. (For instance, it is pertinent to the study of the orbits of dynamical systems.) The evidence for this classification picture is indeed strong, but much work is still needed. It may be too early to evaluate this discovery fully. Earlier classification schemes, well recognized to be important---indeed, mine should only be placed beside them for conceptual comparison!---, are the Linnaeus classification of biological species, the Mendeleev classification of chemical elements, and the classification of subatomic particles in elementary particle physics. (More recently, the Linnaeus classification has virtually given way to that of Watson and Crick!) A mathematical analogue, perhaps, is the classification of finite simple groups. (While the justification of this takes several thousand pages, that much has already been written concerning my conjecture.) My classification scheme (based necessarily on the mathematical notion of a functor, owing to the complexity of the objects considered) has quite new features. The objects mentioned are so complicated that it is not possible to label them with a simple label, in such a way that the label is the same for two objects that are essentially the same. One can only label the objects with other, simpler, objects, in such a way that if two objects are essentially the same, then also the labelling objects are essentially the same ("isomorphic"). I did this forty years ago for an important class of objects (AF C*-algebras---a special kind of amenable C*-algebra). It was fifteen years before I realized that further steps were possible. In the twenty-five years since then, many people have contributed to what has become known as the Elliott program (for the classification of amenable C*-algebras). Great progress has taken place recently (concerning the simple, unital, especially well behaved case), but development continues to snowball. The next five years will be exciting for the Elliott program. (The non-simple, the non-unital, and the not especially well behaved cases---the last in a quite precise sense---have revealed tantalizing challenges.)