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

We introduce the index N (w₁, w₂) 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 w of a two-dimensional interacting electronic system which is symmetric under U(1) charge transformation may be written as the index N (w, w_D), where w_D is obtained from w 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 N .