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 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. [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).

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