# The Blackadar-Handelman conjectures

Von | November 1, 2021

In ​[1]​, Blackadar and Handelman made two conjectures:

Conjecture 1: (Below Theorem I.2.4 in ​[1]​) Let be a unital C*-algebra. Then the set of lower-semicontinuous dimension functions is dense in the set of dimension functions.

Conjecture 2: (Below Theorem II.4.4 in ​[1]​) Let be a unital C*-algebra. Then the compact, convex set is a Choquet simplex.

Here, a dimension function on a unital C*-algebra is a map that associates to every matrix over a positive number and satisfying properties that generalize the classical properties of the rank of complex matrices:

• If and are orthogonal matrices (that is, ), then .
• If is Cuntz-dominated by (that is, there exist sequences and such that ), then .
• .

Such a dimension function is said to be lower-semicontinuous if it is lower-semicontinuous with respect to the norm-topology, that is, whenever is a sequence in converging to some , then . We equip with the topology of pointwise convergence. This gives the structure of a compact, convex set. We note that the subset is usually not closed (indeed, the conjecture is that it is dense). There is another (natural) topology on giving it the structure of a compact, convex set that is even a Choquet simplex. Here, a compact, convex set is a Choquet simplex if the set of continuous, affine functions satisfies Riesz interpolation, that is, given satisfying for there exists such that for . Choquet simplices have the property that every element can be represented in a unique way by a boundary measure. Important examples of Choquet simplices are the Bauer simplices: Given a compact, Hausdorff space , the set of positive, Borel probability measures on is a Choquet simplex with boundary , and .

If is a commutative C*-algebra, then naturally corresponds to the set of finitely-additive probability measures on , while naturally corresponds to the set of (-additive) probability measures on . For a general C*-algebra , we therefore consider the dimension functions on as „noncommutative, finitely-additive probability measures“, and similarly are the „noncommutative probability measures“ on . It is easy to see that the probability measures on a compact, Hausdorff space are dense in the set of finitely-additive probability measures (see the proof of Theorem I.2.4 in ​[1]​), and it is a classical result that the finitely-additive probability measures on a compact, Hausdorff space form a Choquet simplex. The Blackadar-Handelman conjectures predict that theses results generalize to the noncommutative setting.

The first Blackadar-Handelman conjecture has been verified in the following cases: if is commutative (Theorem I.2.4 in ​[1]​); if is simple, exact, stably finite and has strict comparison of positive elements (Theorem B and 6.4, and Remark 6.5 in ​​[2]​​); if has finite radius of comparison (Theorem 3.3 in ​[3]​).

The second Blackadar-Handelman conjecture has been verified in the following cases: if is commutative; if is simple, exact, stably finite and -stable (Theorem B in ​​[2]​); if has real rank zero and stable rank one (Corollary 4.4 in ​​[4]​); in ​​[5]​, the assumption of real rank zero was removed from the result in ​​[4]​, thus verifying the second Blackadar-Handelman conjecture for all C*-algebras of stable rank one.

With view to the results in ​[5]​ it is natural to ask if the first Blackadar-Handelmann conjecture can be verified for all C*-algebras of stable rank one.

1. [1]
B. Blackadar, D. Handelman, Dimension functions and traces on C∗-algebras, Journal of Functional Analysis. (1982) 297–340. https://doi.org/10.1016/0022-1236(82)90009-x.
2. [2]
N.P. Brown, F. Perera, A.S. Toms, The Cuntz semigroup, the Elliott conjecture, and dimension functions on C*-algebras, Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal). (2008). https://doi.org/10.1515/crelle.2008.062.
3. [3]
K. De Silva, A note on two Conjectures on Dimension funcitons of C*-algebras, ArXiv. (2016) 1601.03475.
4. [4]
F. Perera, The Structure of Positive Elements for C*-Algebras with  Real Rank Zero, Int. J. Math. (1997) 383–405. https://doi.org/10.1142/s0129167x97000196.
5. [5]
R. Antoine, F. Perera, L. Robert, H. Thiel, C*-algebras of stable rank one and their Cuntz semigroups, Duke Math. J. (2022) (to appear).