Operator Algebras and Applications

这项拟议的研究的目的是证实一个惊人的现象。我已经发现，一大类相当复杂的数学对象(几乎所有在希尔伯特空间中自然产生的算子的范数闭代数，包括那些对数学物理感兴趣的，以及在其中考虑复杂系统的其他数学分支，如动力系统理论和叶理论)显然可以用非常简单的数据来描述-现在通常被称为埃利奥特不变量。*这是意想不到的，但证据是压倒性的。这一发现对所涉及的对象类别(可服从C*-代数的类别)都具有深远的影响，而且似乎对这些对象所在的相关数学和物理领域也是如此。(例如，它与动力系统轨道的研究有关。)*这种分类的证据确实很充分，但仍有许多工作要做。*现在全面评估这一发现可能还为时过早。早期的分类方案被公认为很重要-事实上，我的分类方案只应放在它们旁边进行概念比较！-是林奈对生物物种的分类，门捷列夫对化学元素的分类，以及基本粒子物理学中的亚原子粒子分类。(最近，林奈的分类实际上已经让位于沃森和克里克的分类！)或许，一个数学类比就是有限单群的分类。*我的分类方案(由于所考虑的对象的复杂性，必须基于函数式的数学概念)具有相当新的特征。所提到的对象是如此复杂，以至于不可能用简单的标签来标记它们，使得对于本质上相同的两个对象的标签是相同的。人们只能用其他更简单的对象来标记对象，如果两个对象本质上相同，那么标记对象也本质上相同(“同构”)。*40年前我对一类重要的对象(AFC*-代数-一种特殊的服从C*-代数)做了这样的标记。15年后，我才意识到有可能采取进一步的措施。在那之后的25年里，许多人为众所周知的Elliott程序(用于依从性C*-代数的分类)做出了贡献。*最近取得了很大的进展(关于简单的、单位的，特别是行为良好的情况)，但发展继续滚雪球。对于埃利奥特项目来说，未来五年将是令人兴奋的。(不简单的、非统一医院的和表现不是特别好的案例-非常准确地说，最后一个案例-揭示了诱人的挑战。)

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.)