Speaker: Kang Li (FAU Erlangen-Nürnberg)

Title: Dimension theories from groupoids to classifiable C^*-algebras, and back again

Time/Date: 4:45-6:15pm, September 9 (Tue.), 2025

Room: 126 Math. Sci. Building (It will be also online. The Zoom link is the same as before. If you don't have one, please ask Kawahigashi.)

Abstract: The motivation comes from the spectacular breakthrough in the Elliott classification program for simple nuclear C^*-algebras: the class of all separable, simple, finite nuclear dimensional C^*-algebras satisfying the UCT is classified by their Elliott invariants. Shortly after, Xin Li proved that those classifiable C^*-algebras have a twisted étale groupoid model (G, Σ). A natural question is which twisted étale groupoid C^*-algebras have finite nuclear dimension. Very recently, Bönicke and I have extended the previous results to show that their nuclear dimensions are bounded by the dynamic asymptotic dimension of the underlying groupoid G and the covering dimension of its unit space G⁰, and are actually independent of Σ. The essential flaw is that dynamic asymptotic dimension cannot be consistent with nuclear dimension for simple C^*-algebras because every simple C^*-algebra with finite nuclear dimension has nuclear dimension either zero or one. Therefore, we (together with Liao and Winter) introduced the so-called diagonal dimension for an inclusion (D ⊆ A) of C^*-algebras. In this talk, I will explain how the diagonal dimension of (C₀(G⁰)⊆ C_r^*(G,Σ)) is indeed consistent with dynamic asymptotic dimension of G and the covering dimension of G⁰. Moreover, we compute the diagonal dimension and the dynamic asymptotic dimension for Xin Li's groupoid model. This is joint work with Christian Bönicke, as well as Zehong Huang and Hang Wang.