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

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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.