A noncommutative version of Farber's topological complexity

Abstract

Topological complexity for spaces was introduced by M. Farber as a minimal number of continuity domains for motion planning algorithms. It turns out that this notion can be extended to the case of not necessarily commutative $C^*$-algebras. Topological complexity for spaces is closely related to the Lusternik-Schnirelmann category, for which we do not know any noncommutative extension, so there is no hope to generalize the known estimation methods, but we are able to evaluate the topological complexity for some very simple examples of noncommutative $C^*$-algebras.