The classification problem for amenable $C^*$-algebras
George A. Elliott (University of Copenhagen and University of Toronto
Recent evidence suggests that it may be possible to classify all naturally arising separable $C^*$-algebras, by means of quite simple invariants. In technical terms, the appropriate class would seem to be the amenable $C^*$-algebras (equivalently, the nuclear ones).
Such a classification would be analogous to the classification by Connes, Haagerup, Krieger, and Takesaki of amenable von Neumann algebras on a separable Hilbert space. One might ask if the von Neumann algebra classification could be used in obtaining a $C^*$-algebra classification (in view of the fact that a $C^*$-algebra is amenable if and only if its bidual is amenable). There would appear to be no direct connection.
Such a classification would of course give nothing new in the commutative case---beyond the Gel\cprime fand-Na\u\i mark description of the $C^*$-algebra in terms of its spectrum. (One cannot expect to have a complete invariant for locally compact Hausdorff spaces, other than the space itself.)
In the case of simple $C^*$-algebras, such a classification would be striking.
It is possible that for stable simple $C^*$-algebras (i.e., for the class of simple $C^*$-algebras tensored with the $C^*$-algebra of compact operators on a separable infinite-dimensional Hilbert space), the invariant may be just as follows:
(i) the $K_0$-group, with its natural pre-order structure;
(ii) the $K_1$-group;
(iii) the space of densely defined, lower semicontinuous, positive traces, with its structure of topological convex cone;
(iv) the natural pairing of the cone of traces with the $K_0$-group.
The non-stable case would involve additional information---but in the unital case presumably just the $K_0$-class of the unit.
As well as the question of the completeness of the invariant (now established for several classes of algebras), there is also the question of its range. In particular (in the simple, stable case), what properties characterize the objects arising as above from separable amenable stable simple $C^*$-algebras?
There are certain properties the invariant is known to have, in the (separable, amenable) stable simple case:
(i) the $K_0$-group is countable (and abelian), and the pre-order structure is simple;
(ii) the $K_1$-group is countable (and abelian);
(iii) the cone of traces is non-zero whenever $K_0$ has a non-positive element, and, when non-zero, has a compact base which is a Choquet simplex;
(iv) the pairing of the cone of traces with $K_0$ gives rise to the cone of all strictly positive additive real-valued functionals on $K_0$, together with 0.
Some evidence suggests that these properties characterize what arises as the invariant in the stable simple case.
One class of stable simple $C^*$-algebras for which the isomorphism theorem is known consists of those obtained as the inductive limits of sequences of finite direct sums of matrix algebras over the commutative algebra $C({\ssf T})$. In this case, one also knows, through work of Thomsen and Villadsen, what arises as the invariant.
Other, related, results are known, as the result of the work of a number of authors. (One should perhaps also mention the earlier work on the approximately finite-dimensional case of Glimm, Dixmier, Bratteli, Effros, Handelman, Shen and the author.)
Another class of (separable, amenable) stable simple $C^*$-algebras for which an isomorphism theorem is known---owing to work of Bratteli, Evans, Kishimoto, H. Lin, N. C. Phillips, R\o rdam, St\o rmer, and the author---may turn out to consist of all purely infinite ones. This is known at the level of the invariant---which in this case is just the two countable abelian groups $K_0$ and $K_1$. (All possible pairs arise.)