The K-homology of nets of C∗-algebras

Let X be a space, intended as a possibly curved space-time, and A a precosheaf of C∗-algebras on X. Motivated by algebraic quantum field theory, we study the Kasparov and Θ-summable K-homology of A interpreting them in terms of the holonomy equivariant K-homology of the associated C∗-dynamical system. This yields a characteristic class for K-homology cycles of A with values in the odd cohomology of X, that we interpret as a generalized statistical dimension.