Groups and Operator Algebras Seminar

Speaker: Kang Li (Friedrich-Alexander-Universität Erlangen-Nürnberg)

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

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 G^0.

The problem 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 (C_0(G^0)⊆ C_r^*(G,Σ)) is indeed consistent with dynamic asymptotic dimension of G and the covering dimension of G^0. Moreover, we compute the diagonal dimension and the dynamic asymptotic dimension for Xin Li’s groupoid model.