非交换维数理论 Non-commutative Dimension Theories

Kang Li 李康  (Friedrich-Alexander-Universitat Erlangen-Nürnberg)

15:15-16:15, April 17, 2024   Mathematics Building 102, Minhang Campus

Abstract:

A C*-algebra is often considered as non-commutative space, which is justified by the natural duality between the category of unital, commutative C*-algebras and the category of compact, Hausdorff spaces. Via this natural duality, we transfer Lebesgue covering dimension on compact, Hausdorff spaces to nuclear dimension on unital, commutative C*-algebras. The notion of nuclear dimension for C*-algebras was first introduced by Winter and Zacharias, and it has come to play a central role in the structure and classification for simple nuclear C*-algebras. Indeed, after several decades of work, one of the major achievements in C*-algebra theory was completed: the classification via the Elliott invariant for non-elementary simple separable unital C*-algebras with finite nuclear dimension that satisfy the universal coefficient theorem. Unfortunately, simple C*-algebras suffer from a phenomenon of dimension reduction: every simple C*-algebra with finite nuclear dimension must have nuclear dimension at most one. In order to overcome this phenomenon, we (together with Liao and Winter) have introduced the notion of diagonal dimension for an inclusion of C*-algebras, where D is a commutative sub-C*-algebra of A so that this new dimension theory generalizes Lebesgue covering dimension of D and nuclear dimension of A simultaneously. In this talk, I will explain its future impact on the classification of simple nuclear C*-algebras and its connection to dynamic asymptotic dimension introduced by Guentner, Willett and Yu.

About the speaker:

李康，丹麦青年数学家，专长于算子代数及其在动力系统，几何群论与表示论中的应用，在IMRN, JFA, Adv Math, Ergodic Theory Dynam. Systems，JNCG发表近30篇论文。

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