Eduard Vilalta (Universitat Autònoma de Barcelona)
The range problem for the Cuntz semigroup of AI-algebras
Abstract:
A C*-algebra A is said to be a (separable) AI-algebra if A is isomorphic to an inductive limit of the form $\lim\limits_n (C[0,1]\otimes M_n)$ with $F_n$ a finite dimensional C*-algebra for every n. Whenever A is unital and commutative, A is isomorphic to C(X) with X an inverse limit of finite disjoint copies of unit intervals.
In this 2-session talk, we will study the range problem for the Cuntz semigroup of AI-algebras. That is, we will study whether or not one can determine a natural set of properties that an abstract Cuntz semigroup must satisfy in order to be isomorphic to the Cuntz semigroup of an AI-algebra.
During the first part of the talk, we will focus on unital commutative AI-algebras. In this case, one is able to solve the range problem for this class, thus giving a list of properties that an abstract Cuntz semigroup S satisfies if and only if S is isomorphic to the Cuntz semigroup of such an algebra. In order to prove this result, we first introduce the notion of almost chainable spaces and prove that a compact metric space X is almost chainable if and only if C(X) is an AI-algebra. We also characterize when S is isomorphic to the Cuntz semigroup of lower-semicontinuous functions $X\to 0,1,\dots,\infty$ for some T1-space X. The results in this first session will appear in [4].
In the second session, we will present a local characterization for the Cuntz semigroup of any AI-algebra resembling Shen’s local characterization of dimension groups[3], later used in the celebrated Effros-Handelman-Shen theorem[2]. One of the key features in the proof of our result will be the notion of Cauchy sequences for Cu-morphisms (with respect to the distance introduced in [1]) and the fact that, under the right assumptions, they have a unique limit; see [5].
[1] Ciuperca, A. and Elliott, G. “A remark on invariants for C*-algebras of stable rank one”, Int. Math. Res. Not. IMRN(2008)
[2] Effros, E. G. and Handelman, D. E. and Shen, C. L. “Dimension groups and their affine representations”, Amer. J. Math.102(1980), 385–407.
[3] Shen, C. L. “On the classification of the ordered groups associated with the approximately finite dimensional C*-algebras” ,Duke Math. J.46(1979), 613–633.
[4] Vilalta, E. “The Cuntz semigroup of unital commutative AI-algebras”, in preparation.
[5] Vilalta, E. “A local characterization for the Cuntz semigroup of AI-algebras”, (preprint) arXiv:2102.13557 [math.OA]