Operator Algebras and Applications

这项研究的目的是验证我最近观察到的一个相当惊人的现象。大量的证据表明，一大类相当复杂的数学对象-实际上所有自然产生的Hilbert空间中的范数闭代数-都是由它们所构成的（包括那些对数学物理感兴趣的人，以及考虑复杂系统的其他数学分支，如动力系统理论和叶理理论）-可以完全用简单的数据来描述-所谓的K理论不变量，有时被称为艾略特不变量 一方面，这是最出乎意料的。(But证据是压倒性的）。另一方面，这一发现有着深远的影响，既对所涉及的对象类（用专业术语来说，顺从的C*-代数类），也很可能对这些对象出现的数学和物理学的相关领域产生影响。(For例如，它与动力系统的研究有关。粗略地说，对于Cantor系统，C*-代数不变量和轨道结构不变量是相同的！） 对于我所提出的分类图，有几种证据确实很有力。然而，要证实我的猜想的真实性，仍然需要投入相当多的资源。 要全面评估这一发现的重要性可能还为时过早。较早的分类方案，公认是重要的，是生物物种的林奈分类，通过几个层次的有序划分，门捷列夫分类的化学元素，通过周期表，和分类的亚原子粒子在基本粒子物理学，作为不同种类的夸克的集合。一个数学上的类比，也许是有限单群的分类。我的分类方案，是必要的基础上的概念，一个函子，由于复杂的对象考虑，有相当新的特点。在我最早的工作之后不久，康纳斯和其他人为同样复杂的一类对象（顺从的冯·诺依曼代数）开发了一个类似的方案，非常成功。

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.