(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 ofalways 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.
Question 4:
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
.