The C0(X)C0(X)-algebra of a net and index theory

Given a connected and locally compact Hausdorff space X with a good base K   we assign, in a functorial way, a C0(X)C0(X)-algebra to any precosheaf of C⁎C⁎-algebras AA defined over K. Afterwards we consider the representation theory and the Kasparov K  -homology of AA, and interpret them in terms, respectively, of the representation theory and the K  -homology of the associated C0(X)C0(X)-algebra. When AA is an observable net over the spacetime X in the sense of algebraic quantum field theory, this yields a geometric description of the recently discovered representations affected by the topology of X.