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

希尔伯特空间上算子代数上有限生成的射影模的研究与代数本身一样长。早在20世纪30年代，Murray和von Neumann就已经使用这些模块的同构类结构来对因子进行类型分类。后来，这些相同的类被用来建立算子代数的k理论的开端。这把我们带到了这个建议的起点：可分可调C*-代数的k理论刚性现象。Elliott在20世纪90年代初提出的猜想是，这些代数应该由它们的巴拿赫代数k理论和迹泛函确定到同构。我们现在知道，这个猜想最大胆的时候是站不住脚的。罗达姆在2003年发现了第一个反例，我（独立地）在稳定有限情况下发现了更多的反例。最后一点指出了对可分离可控制C*-代数进行分类的一个重要缺点：在C*-代数上有限生成的射影模所携带的信息量与它的可数生成的希尔伯特模所携带的信息量之间存在很大的差距（给定C*-代数上的Banach模在同一代数上具有内积取值）。例如，有限生成模对紧可缩Hausdorff空间X上的连续复值函数的代数本质上是零，而Hilbert模则完全恢复X。

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.