Renormalized traces as a geometric tool (Q2751076)

The paper is an extensive treatment of renormalized traces. The goal is to extend the concepts involving the trace (divergence of one form, the Ricci curvature, the Laplacian) from finite-dimensional manifolds to infinite-dimensional cases, namely to a class of Frechet, resp. Hilbert manifolds and vector bundles. In the first step the trace on the algebra of matrices is generalized to the algebra of classical pseudodifferential operators on some closed manifolds. These traces are built on finite parts of otherwise divergent expressions. The obstruction to extend concepts and results of Riemannian geometry or classical constructions in the complex setting (e.g. Chern-Weil construction) in a straightforward manner are described in terms of Wodzicki residues of classical pseudodifferential operators. The cases when these obstructions vanish are discussed.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00053].