The cuntz semigroup and the structure of c*-algebras: dynamics and classification

cuntz 半群和 c* 代数的结构：动力学和分类

• 批准号: 311669-2010
• 负责人: Toms, Andrew
• 金额: $ 1.42万
• 依托单位: York University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2010
• 资助国家: 加拿大
• 起止时间: 2010-01-01 至 2011-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=457090
• 关键词: cuntz; semigroup; structure; algebras; dynamics

Finitely generated projective modules over algebras of operators on Hilbert space have been studied for as long as the algebras themselves. Already in the 1930s Murray and von Neumann used the structure of isomorphism classes of these modules to give their type classification of factors. Later, these same classes were used to establish the beginnings of K-theory for operator algebras. This brings us to the starting point of this proposal: the phenomenon of K-theoretic rigidity for separable amenable C*-algebras. The conjecture, put forward by Elliott in the early 1990s, is that the these algebras should be determined up to isomorphism by their Banach algebra K-theory and tracial functionals. We know now that the conjecture does not hold at its most daring. Rordam found the first counterexample in 2003, and I discovered (independently) further counterexamples in the stably finite case. The last of these fingers an important shortcoming in the effort to classify separable amenable C*-algebras: there is a large gap between the amount of information carried by the finitely generated projective modules over a C*-algebra and that carried by its countably generated Hilbert modules (Banach modules over the given C*-algebra which are equipped with an inner product taking values in the same algebra). The finitely generated modules, for instance, say essentially nil about the algebra of continuous complex-valued functions on a compact contractible Hausdorff space X, while the Hilbert modules recover X completely.

希尔伯特空间上算子代数上的直接生成投射模的研究时间与代数本身一样长。 早在20世纪30年代默里和冯诺依曼使用的结构同构类的这些模块给他们的类型分类的因素。 后来，这些相同的类被用来建立算子代数的K-理论的开端。 这就把我们带到了这个提议的出发点：可分顺从C*-代数的K-理论刚性现象。 由Elliott在90年代初提出的猜想是，这些代数应该由它们的Banach代数K-理论和迹泛函决定到同构。 我们现在知道，这个猜想在最大胆的时候并不成立。 Rordam在2003年发现了第一个反例，我（独立地）发现了稳定有限情形下的更多反例。 最后这些手指的一个重要缺点是在分类可分顺从C*-代数的努力：有一个很大的差距，所携带的信息量之间的cumber-generated投射模在一个C*-代数，并进行其可数生成希尔伯特模（巴拿赫模在给定的C*-代数，配备了一个内积在同一个代数的值）。 例如，希尔伯特生成模对于紧致可收缩豪斯多夫空间X上的连续复值函数的代数本质上是零，而希尔伯特模则完全恢复了X。