Operator Algebras and Applications

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The objective of the proposed research is to verify a rather striking phenomenon that I have recently observed. Considerable evidence has now been assembled that a large class of quite complicated mathematical objects---virtually all naturally arising norm-closed algebras 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 be completely described in terms of simple data---the so-called K-theory invariant, sometimes now referred to as the Elliott invariant. On the one hand, this is most unexpected. (But the evidence is overwhelming.) On the other hand, this discovery has far-reaching implications, both for the class of objects involved (in technical terms, 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 dynamical systems. Roughly speaking, for Cantor systems, the C*-algebra invariants and the orbit structure invariants are the same!) The evidence, of several kinds, for the classification picture I have proposed, is indeed strong. To establish the truth of my conjecture, however, will still require a considerable investment of resources. It may be too early for the importance of this discovery to be fully assessed. Earlier classification schemes, well recognized to be important, are the Linnaeus classification of biological species, by means of several levels of orderly division, the Mendeleev classification of chemical elements, by means of the periodic table, and the classification of subatomic particles in elementary particle physics, as assemblages of different kinds of quarks. A mathematical analogue, perhaps, is the classification of finite simple groups. My classification scheme, being based necessarily on the concept of a functor, owing to the complexity of the objects considered, has quite new features. A similar scheme, developed by Connes and others shortly after my earliest work, for an equally complicated class of objects (the amenable von Neumann algebras), was very successful.