Speaker: Hannes Thiel (TU Dresden)

Title: The generator rank of C*-algebras

Time/Date: 4:45-6:15pm, June 8 (Tue.), 2021

Room: Online

Abstract: The generator problem asks to determine for a given C^*-algebra its minimal number of generators. In particular, one wants to know if every separable, simple C^*-algebra is generated by a single element. The generator problem was originally asked for von Neumann algebras, and Kadison included it as Nr. 14 of his famous list of 20 "Problems on von Neumann algebras". The problem remains open, most notably for the reduced free group C^*-algebras and the free group factors.

The generator rank is a stable quantification of the generator problem. Instead of asking if a C^*-algebra is generated by k elements, the generator rank records whether the generating k-tuples are dense. It turns out that this invariant has good permanence properties and it can be computed in a number of interesting cases.

In this talk, I will first give an overview on the generator problem and then present some recent results on computations of the generator rank. Most interestingly, we will see a strong solution to the generator problem for separable, simple, classifiable C^*-algebras: They are not merely singly generated but they contain a dense set of generators.