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

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.