As mentioned in the abstract, the notion of spectral localizers is extended to pairings with semifinite spectral triples. By the spectral flow argument, the index pairing of semifinite is shown to be equal of the signature of the spectral localizer. As an application, derived is the formula for weak invariants of topological insulators, providing a new approach in numerical evaluation.
Table of Contents: 1. Overview 2. Semifinite spectral flow and Signature 3. Spectral localizer 4. Constancy of the signature 5. Proof of the odd index formula 6. Proof of the even indenx formula 7. Application to topological insulators References
Reviewer’s remark: A semifinite spectral triple means a tuple \((N, A, D)\) of a semifinite von Neumann algebra \(N\) with a semifinite faithful normal trace, a star subalgebra \(A\) of \(N\), and a self-adjoint operator \(D\) affiliated to \(N\), these which satisfy a few of differential and spectral conditions. The triple is even if there is a self-adjoint unitary to make the \(2\) by \(2\) matrix argument like a spin, and odd otherwise. For an odd semifinite spectral triple, by doubling, the spectral localizer is defined to be a \(2\times 2\) matrix with \(k D\) and \(-kD\) on the diagonal and \(a\) and \(a^*\) on the off diagonal for \(k\) positive and \(a\in A^{\sim}\) as an operator affiliated to \(N\) tensored with the matrix algebra \(M_2(\mathbb C)\). In the even case, it is within \(N\) plus \(D\).