George A. Elliott, University of Toronto

How amenable are amenable operator algebras?

Amenability is a property both of von Neumann algebras and of C*-algebras, with an appropriate definition in each case. A discrete group is amenable in the classical sense if, and only if, all the C*-algebras and von Neumann algebras generated by it are amenable. (For a more general locally compact group the situation is a bit more subtle.) There are many equivalent definitions of amenability for both C*-algebras and von Neumann algebras. In addition, a C*-algebra is amenable if, and only if, all the von Neumann algebras generated by it (in representations) are amenable. Every amenable von Neumann algebra is generated by some amenable C*-algebra. Amazingly, even though the classification of either amenable von Neumann algebras or amenable C*-algebras might, on the face of it, even with appropriate countability assumptions, threaten to be just as complicated as that of amenable groups---which is surely hopeless---even in the abelian case!---, in fact, amenable von Neumann algebras are completely classified by a simple invariant, and recently (but with a long history) there is an analogous result for amenable C*-algebras---with a certain additional well-behavedness property that is quite simple to state. Most (or at least many!) naturally occurring C*-algebras or von Neumann algebras are amenable, just as is true for groups. There are also many, many examples of amenable C*-algebras which are well enough behaved to fit into the recent classification. Indeed, the class in question is just just those simple (separable, amenable) C*-algebras that absorb tensorially a certain one of them, which in a strong sense is just a souped-up version of the complex numbers. (There is an additional technical assumption, which however may hold automatically.) Since this funny algebra of complex numbers absorbs itself, the tensor product of an arbitrary simple separable amenable C*-algebra with it also absorbs it---and is therefore in the classifiable class.