A noncommutative version of Farber's topological complexity

TL;DR

This paper extends the concept of topological complexity from topological spaces to noncommutative C*-algebras, providing explicit calculations for simple cases and highlighting the challenges in generalizing classical estimation methods.

Contribution

It introduces a noncommutative version of Farber's topological complexity and evaluates it for specific simple noncommutative C*-algebras, addressing a gap in the theory.

Findings

Topological complexity can be extended to noncommutative C*-algebras.

Explicit calculations are provided for simple noncommutative examples.

Classical estimation methods do not directly generalize to the noncommutative setting.

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.