(Based on the talk „Thoughts on the classification problem for amenable C*-algebras“ of George Elliott, 30. June 2020, at the Zagreb Workshop on Operator Theory.)
Let us consider the class of unital, separable, simple, nuclear C*-algebras. The Toms-Winter conjecture predicts that for such an algebra , the following conditions are equivalent:
- has finite nuclear dimension.
- is -stable, that is, where denote the Jiang-Su algebra.
- has strict comparison of positive elements, which means that the Cuntz semigroup of is almost unperforated: if and satisfy for some , then .
As of today, it is known that (1) and (2) are equivalent and that (2) implies (3). Further, it is known that (3) implies (2) under certain additional assumptions.
As long as the Toms-Winter conjecture is not completely verified, the second condition seems most natural, and we say that is regular if it is -stable. By the spectacular recent breakthrough in the Elliott classification program, it is known that every regular C*-algebras that satisfy the Universal Coefficient Theorem (UCT) are classified by the Elliott invariant (-theory and tracial data).
We have an obvious dichotomy: a unital, separable, simple, nuclear C*-algebra is either regular or nonregular. It is easy to find regular algebra. Indeed, since the Jiang-Su algebra is self-absorbing, that is, , for every the algebra is automatically regular. Examples of nonregular C*-algebras were constructed by Villadsen, Toms and Rørdam.
Question 1:Are the regular or the nonregular C*-algebras generic?
A regular C*-algebra is either purely infinite or stably finite. A simple, unital, purely infinite C*-algebra has infinite stable rank. Rørdam showed that a stably finite, regular C*-algebra has stable rank one. Thus, a simple C*-algebra with finite stable rank is nonregular. Villadsen showed that for every there exists a simple AH-algebra with stable rank . He also showed that his algebra with stable rank has either real rank or .
Question 2:Does there exist a nonregular C*-algebra of real rank zero?
Question 3:What is the relation between the stable rank and real rank of a stably finite, simple (nuclear) C*-algebra? Is the real rank of always either or ? Is this connected to the number of tracial states on ?
Elliott suggests that should correspond to a unique (or very few) tracial state on , while should correspond to a large tracial simplex of . It is known that for every C*-algebra . Moreover, if is a compact, Hausdorff space, then and . Thus, the stable rank of a commutative C*-algebra is roughly half of its real rank.
Can the recent classification of regular (separable, simple, nuclear) C*-algebras be extended to include (some) nonregular C*-algebras by using a stronger invariant?
A natural candidate for a stronger invariant would be the Cuntz semigroup. The Cuntz semigroup encodes the -group, the simplex of tracial states, and the pairing between traces and . However, it does not encode the -group. One should therefore consider classification by the pair .
Elliott suggests to consider the following testcase: Let and be simple inductive limits of matrix algebras over the Hilbert cube. Then , and the question becomes: Are and isomorphic whenever their Cuntz semigroups are isomorphic? If , and if is regular, then the Cuntz semigroup of is `regular‘ (in the sense of Winter) and it follows that both algebras are regular. Then, since and also satisfy the UCT, one can indeed deduce from the classification result that .