A central aim in the mathematics of topological phases of matter is to identify macroscopic observables with global topological indices and, when possible, show these indices are complete invariants. For the noninteracting quantum Hall effect, e.g., the observable is the Hall conductance and the index is the Chern number or a Fredholm index. We focus on the interacting setting, where formulating the correct infinite-volume topological index has been elusive. We define a discrete "index of a pair of pure states" on an abstract C*-algebra, analogous to the Avron–Seiler–Simon "index of a pair of projections" in Hilbert space, and apply it to the interacting quantum Hall effect, modeled on the CAR algebra over ℓ²(ℤ²). This index is quantized and locally constant under appropriate perturbations, providing a deformation-invariant characterization of the quantum Hall phase. We also discuss ℤ₂-valued variants for different symmetry classes.