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

核 C* 代数和（非交换）动力系统的分类

• 批准号: 0101060
• 负责人: Cornel Pasnicu
• 金额: $ 7.78万
• 依托单位: University of Puerto Rico
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-01 至 2005-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0101060&HistoricalAwards=false
• 关键词: Classification; Nuclear; algebras; Noncommutative; Dynamical

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).

Pasnicu代数（分别为AH代数）是作为子代数的有限直和的归纳极限而产生的C ~*-代数（分别为角子代数），交换C*-代数。一个C*-如果每个理想都是由投影生成的，则称一个代数具有理想性质.本文提出了一个具有理想性质的核ASH代数的分类，并对AH代数进行了分类具有理想性质的C ~*-代数。他还提出了一个猜想，该猜想指出，“许多”具有理想性质的核可分C ~*-代数是一个有单位的交换C ~*-代数或一个AF代数与整数的交叉积，是上述类中的ASH代数。该项目与Elliott的可分C ~*-代数的分类程序有关，核C*-代数和一个Effros问题，并且可能在算子代数，遍历理论，（非交换）动力系统的研究和几何学中产生影响。C*-代数可以被认为是具有有趣的代数和拓扑结构的无限矩阵的集合。 C*-代数有重要的应用，以其他部分的数学（几何，拓扑，遍历理论），部分物理（量子力学和统计力学）或其他科学（结构的DNA和其他分子）。一个完整的分类（“枚举”）一类特殊的运营商代数，称为顺从冯诺依曼代数，是由康纳斯在他的菲尔茨奖获奖作品。 这个项目有两个主要目标。一个是分类（“枚举”）具有理想性质的顺从（核）C*-代数的大类（一个有趣的技术条件）由一个特定的结构定义二是证明了许多具有理想性质的顺从C ~*-代数是由一种完全不同的自然构造而产生的（“交叉产品”）实际上属于上述类别之一（“归纳极限”），研究人员建议分类（“枚举”）。这个项目可能会有重要的影响，在几个数学领域，包括算子代数，动力系统，几何，也在一些领域以外的数学（如量子物理）。