The spectral localiser via E-theory

The numerical invariant of a topological insulator is given by the index of a Fredholm operator constructed from the Hamiltonian. Equivalently, this is the index pairing between the K-theory class of the Hamiltonian and the spectral triple given by the position operator. Hermann Schulz-Baldes and Terry Loring have introduced the spectral localiser as a means to compute the index pairing. These are invertible matrices supported on a finite-dimensional spectral subspace of the spectral triple. In the odd case, we interpret the spectral localiser as an index pairings in E-theory. Namely, the image of an odd K-theory class under the asymptotic morphism generated by an odd spectral triple. This allows for speaking of spectral truncation of index pairings in the framework of bivariant K-theory.

This is based on joint work with Bram Mesland.