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

Hilbert空间上算子代数上的有限生成投射模的研究，早在代数本身的研究中就已经有了。早在20世纪30年代，Murray和von Neumann就利用这些模的同构类的结构给出了它们的因子的类型分类。后来，这些相同的类被用来建立算子代数的K-理论的开端。这就把我们带到了这个建议的起点：可分可依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Banach代数K-理论和迹泛函确定为同构。我们现在知道，这个猜想并不是最大胆的。罗丹在2003年发现了第一个反例，我(独立地)在稳定有限的情况下发现了更多的反例。在对可分的顺从C*-代数进行分类时，最后一个手指是一个重要的缺点：在C*-代数上的有限生成投射模和它的可数生成的Hilbert模(给定的C*-代数上的Banach模具有取值于同一代数的内积)所承载的信息量之间存在着很大的差距。例如，有限生成的模表示紧致可压缩Hausdorff空间X上的连续复值函数的代数基本上为零，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.