The index of a pair of pure states and the interacting integer quantum Hall effect

We introduce the index $\mathcal{N}({\omega }_1,{\omega }_2)$ of a pair of pure states on a unital C*-algebra, which is a generalization of the notion of the index of a pair of projections on a Hilbert space. We then show that the Hall conductance associated with an invertible state $\omega$ of a two-dimensional interacting electronic system which is symmetric under $U(1)$ charge transformation may be written as the index $\mathcal{N}(\omega ,{\omega }_D)$, where ${\omega }_D$ is obtained from $\omega$ by inserting a unit of magnetic flux. This exhibits the integrality and continuity properties of the Hall conductance in the context of general topological features of $\mathcal{N}$.