Noncommutative Local Systems

Abstract

Gelfand - Na\u{i}mark theorem supplies a one to one correspondence between commutative $C^*$-algebras and locally compact Hausdorff spaces. So any noncommutative $C^*$-algebra can be regarded as a generalization of a topological space. Generalizations of several topological invariants may be defined by algebraic methods. For example Serre Swan theorem states that complex topological $K$-theory coincides with $K$-theory of $C^*$-algebras. This article is concerned with generalization of local systems. The classical construction of local system implies an existence of a path groupoid. However the noncommutative geometry does not contain this object. There is a construction of local system which uses covering projections. Otherwise a classical (commutative) notion of a covering projection has a noncommutative generalization. A generalization of noncommutative covering projections supplies a generalization of local systems.