Fundamental group for $C^*$-algebras

By Gelfand's theorem, there is a correspondence between commutative $C^*$-algebras and locally compact Hausdorff topological spaces. Because of this, (noncommutative) $C^*$-algebras can be interpreted as generalizations of topological spaces.
The talk is devoted to generalizing the concept of a fundamental group. As a prototype, we use the construction of a fundamental group in algebraic geometry. The talk on this topic was first given in 2004, but it contained some errors. They were fixed in 2014, but the resulting constructions were too complicated for a seminar talk. Recently we found an elegant construction which can be explained in one lecture. For details, see https://arxiv.org/abs/1904.13130 .