The Blackadar-Handelman conjectures

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

Conjecture 1: (Below Theorem I.2.4 in [1]) Let $A$ be a unital C*-algebra. Then the set $\mathrm{LDF}(A)$ of lower-semicontinuous dimension functions is dense in the set $\mathrm{DF}(A)$ of dimension functions.

Conjecture 2: (Below Theorem II.4.4 in [1]) Let $A$ be a unital C*-algebra. Then the compact, convex set $\mathrm{DF}(A)$ is a Choquet simplex.

Here, a dimension function on a unital C*-algebra $A$ is a map $d\colon M_\infty(A)\to[0,\infty]$ that associates to every matrix over $A$ a positive number and satisfying properties that generalize the classical properties of the rank of complex matrices:

If $a$ and $b$ are orthogonal matrices (that is, $ab=a^*b=ab^*=a^*b^*=0$ ), then $d(a+b)=d(a)+d(b)$ .
If $a$ is Cuntz-dominated by $b$ (that is, there exist sequences $(r_n)_n$ and $(s_n)_n$ such that $\|a-r_nbs_n\|\to 0$ ), then $d(a)\leq d(b)$ .
$d(1)=1$ .

Such a dimension function is said to be lower-semicontinuous if it is lower-semicontinuous with respect to the norm-topology, that is, whenever $(a_n)_n$ is a sequence in $M_\infty(A)$ converging to some $a$ , then $d(a)\leq\liminf_n d(a_n)$ . We equip $\mathrm{DF}(A)$ with the topology of pointwise convergence. This gives $\mathrm{DF}(A)$ the structure of a compact, convex set. We note that the subset $\mathrm{LDF}(A)\subseteq\mathrm{DF}(A)$ is usually not closed (indeed, the conjecture is that it is dense). There is another (natural) topology on $\mathrm{LDF}(A)$ giving it the structure of a compact, convex set that is even a Choquet simplex. Here, a compact, convex $K$ set is a Choquet simplex if the set $\mathrm{Aff}(K)$ of continuous, affine functions $K\to\mathbb{R}$ satisfies Riesz interpolation, that is, given $f_1,f_2,g_1,g_2\in\mathrm{Aff}(K)$ satisfying $f_j\leq g_k$ for $j,k\in\{1,2\}$ there exists $h\in\mathrm{Aff}(K)$ such that $f_j\leq h\leq g_k$ for $j,k\in\{1,2\}$ . 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 $X$ , the set $M_1(X)$ of positive, Borel probability measures on $X$ is a Choquet simplex with boundary $\partial_e M_1(X)\cong X$ , and $\mathrm{Aff}(M_1(X))\cong C(X,\mathbb{R})$ .

If $A=C(X)$ is a commutative C*-algebra, then $\mathrm{DF}(C(X))$ naturally corresponds to the set of finitely-additive probability measures on $X$ , while $\mathrm{LDF}(C(X))$ naturally corresponds to the set of ( $\sigma$ -additive) probability measures on $X$ . For a general C*-algebra $A$ , we therefore consider the dimension functions on $A$ as „noncommutative, finitely-additive probability measures“, and similarly $\mathrm{LDF}(A)$ are the „noncommutative probability measures“ on $A$ . 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 $A$ is commutative (Theorem I.2.4 in [1]); if $A$ is simple, exact, stably finite and has strict comparison of positive elements (Theorem B and 6.4, and Remark 6.5 in [2]); if $A$ has finite radius of comparison (Theorem 3.3 in [3]).

The second Blackadar-Handelman conjecture has been verified in the following cases: if $A$ is commutative; if $A$ is simple, exact, stably finite and $\mathcal{Z}$ -stable (Theorem B in [2]); if $A$ 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]
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]
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]
K. De Silva, A note on two Conjectures on Dimension funcitons of C*-algebras, ArXiv. (2016) 1601.03475.
[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]
R. Antoine, F. Perera, L. Robert, H. Thiel, C*-algebras of stable rank one and their Cuntz semigroups, Duke Math. J. (2022) (to appear).

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