Traces of holomorphic families of operators on the noncommutative torus and on Hilbert modules

TL;DR

This paper explores traces of holomorphic operator families on the noncommutative torus and Hilbert modules, linking residues to geometric data and extending classical index theorems to noncommutative settings.

Contribution

It extends the theory of traces of holomorphic pseudodifferential operators to noncommutative geometries, connecting residues with geometric invariants and revisiting classical index results.

Findings

Residues at poles relate to geometric invariants.

Extended Wodzicki residues encode local geometric information.

Reinterpretation of Atiyah's L^2-index theorem in noncommutative geometry.

Abstract

We revisit traces of holomorphic families of pseudodifferential operators on a closed manifold in view of geometric applications. We then transpose the corresponding analytic constructions to two different geometric frameworks; the noncommutative torus and Hilbert modules. These traces are meromorphic functions whose residues at the poles as well as the constant term of the Laurent expansion at zero (the latter when the family at zero is a differential operator) can be expressed in terms of Wodzicki residues and extended Wodzicki residues involving logarithmic operators. They are therefore local and contain geometric information. For holomorphic families leading to zeta regularised traces, they relate to the heat-kernel asymptotic coefficients via an inverse Mellin mapping theorem. We revisit Atiyah's L^2-index theorem by means of the (extended) Wodzicki residue and interpret the scalar curvature on the noncommutative two torus as an (extended) Wodzicki residue.