Mathematical Sciences: On the Classification of the Simple, Stably Finite, Separable, Amenable C-Algebras

ASH代数（或AH代数）是由可交换C*-代数上的矩阵代数的子代数（或角子代数）的直和的归纳极限而产生的C*-代数。这些代数在分类可分可调C*-代数的Elliott程序中起着重要的作用。研究人员提出构造具有某些一般性质的C*-代数的ASH（或AH）归纳极限分解，特别是证明任何实秩为0且稳定秩为1的简单可分可适应C*-代数是AH代数的猜想。他们还建议根据k理论和迹迹状态空间不变量（在特殊情况下，由Elliott和包括研究人员在内的其他人使用）对任意简单ASH代数进行分类。量子物理学中从有限自由度到无限自由度的过渡导致了某些无限维代数的数学理论，称为算子代数。一个特殊的算子代数的完整枚举（或分类），称为（可调整的）von Neumann代数，甚至是由A. cones在他1975年获得菲尔兹奖的工作中提出的。（菲尔兹奖相当于数学界的诺贝尔奖。）最近，v·f·r·琼斯应用冯·诺依曼代数理论来研究普通结。这项工作也获得了菲尔兹奖，它揭示了量子物理学和DNA结构。研究人员建议研究另一类重要的算子代数的完整枚举（或分类），称为可调节C*-代数。这种C*代数的成功分类对于我们理解量子物理的无限维世界至关重要。

DMS-9622250 PI: Gong The ASH algebras (or AH algebras, respectively) are the C*-algebras arising as inductive limits of direct sums of subalgebras (or corner subalgebras, respectively) of matrix algebras over commutative C*-algebras. These algebras play an essential role in the Elliott program for the classification of separable amenable C*-algebras. The investigators propose to construct ASH (or AH) inductive limit decompositions for C*-algebras with certain general properties --- and, in particular, to prove the conjecture that any simple separable amenable C*-algebra of real rank zero and stable rank one is an AH algebra. They also propose to classify arbitrary simple ASH algebras in terms of the K-theory and tracial state space invariants (used, in special cases, by Elliott and others including the investigators). The passage from a finite to an infinite number of degrees of freedom in quantum physics led to the mathematical theory of certain infinite dimensional algebras, called operator algebras. A complete enumeration (or classification) of a special class of operator algebras, called (amenable) von Neumann algebras, was eiven by A. Connes in his Fields Medal winning work of 1975. (The Fields Medal is the equivalent of the Nobel Prize in mathematics.) More recently, V. F. R. Jones has applied the theory of von Neumann algebras to study ordinary knots. This work, which also won a Fields Medal, has shed light on quantum physics, and on the structure of DNA. The investigators propose to work on a complete enumeration (or classification) of another important class of operator algebras, called amenable C*-algebras. The successful classification of such C*-algebras will be essential for us to understand the infinite dimensional world of quantum physics.