Classification of Nuclear C*-algebras and (Noncommutative) Dynamical Systems

ASH代数（分别为AH代数）是C*-代数，由矩阵代数的子代数（分别为角子代数）在一元可交换C*-代数上的有限直和的归纳极限产生。如果每个理想都是由投影产生的，那么C*-代数就具有理想性质。研究者提出对一类具有理想性质的核ASH代数进行分类，并对具有理想性质的核ASH代数进行分类。他还提出了一个猜想，该猜想指出，具有理想性质的“许多”核可分C*-代数是整数的可交换单C*-代数或AF代数的叉积，是上述类中的ASH代数。该项目与Elliott的可分离核C*-代数分类计划和Effros问题有关，可能对算子代数、遍历理论、（非交换）动力系统研究和几何都有影响。C*-代数可以被认为是具有有趣的代数和拓扑结构的无限数矩阵的集合。C*代数在数学的其他部分（几何、拓扑、遍历理论）、物理学的某些部分（量子力学和统计力学）或其他科学（DNA和其他分子的结构）中有着重要的应用。一个完整的分类（“枚举”）的一个特殊类别的算子代数，称为可服从冯·诺伊曼代数，是由Connes在他的菲尔兹奖获奖作品中给出的。这个项目有两个主要目标。一种是分类（“列举”）具有理想性质（一个有趣的技术条件）的大类可适应（核）C*-代数，这些代数是由一个特定的构造（“归纳极限”）定义的。另一个是证明许多具有由完全不同的自然构造（“交叉积”）产生的理想性质的可服从的C*-代数实际上属于研究者提议分类（“枚举”）的上述类（“归纳极限”）之一。这个项目可能会对几个数学领域产生重要影响，包括算子代数、动力系统、几何以及数学以外的一些领域（例如量子物理）。

AbstractPasnicuThe ASH algebras (respectively AH algebras) are C*-algebras arising as inductive limits of finite direct sums of subalgebras (respectively corner subalgebras) of matrix algebras over unital,commutative C*-algebras.A C*-algebra is said to have the ideal property if each ideal is generated by projections.The investigator proposes to classify a large class of nuclear ASH algebras with the ideal property and also to classify the AH algebras with the ideal property.He also proposes to work on a conjecture which states that "many" nuclear,separable C*-algebras with the ideal property which are the crossed product of a unital,commutative C*-algebra or of an AF algebra by the integers is an ASH algebra in the above class.This project is related to Elliott's program of the classification of the separable,nuclear C*-algebras and to a problem of Effros and could have an impact in operator algebras but also in ergodic theory,in the study of the (noncommutative) dynamical systems and in geometry.C*-algebras could be thought as collections of infinite matrices of numbers endowed with an interesting algebraic and topological structure. The C*-algebras have significant applications to other parts of mathematics (geometry,topology,ergodic theory),to parts of physics (quantum mechanics and statistical mechanics) or to other sciences (the structure of DNA and other molecules).A complete classification ("enumeration") of a special class of operator algebras,called amenable von Neumann algebras,was given by Connes in his Fields Medal winning work. This project has two main goals.One is to classify ("enumerate") large classes of amenable (nuclear) C*-algebras with the ideal property (an interesting technical condition) which are defined by a particular construction ("inductive limits").The other one is to show that many amenable C*-algebras with the ideal property arising from a completely different and natural construction ("crossed products") belong in fact to one of the above classes (of "inductive limits") that the investigator proposes to classify ("enumerate").This project could have an important impact in several mathematical fields including operator algebras, dynamical systems,geometry and also in some domains outside mathematics (e.g. in quantum physics).