Extensions and renormalized traces (Q411347)

For an extension \(0\to\mathcal B\to\mathcal E\to\mathcal A\to 0\) of algebras over \(\mathbb C\) let \(E:HP^0(\mathcal B)\to HP^1(\mathcal A)\) denote the connecting morphism in periodic cyclic cohomology. The goal of the paper is to present an explicit construction for \(E\). The method is related to the renormalization procedure introduced by the author [J. Geom. Phys. 60, No. 10, 1441--1473 (2010; Zbl 1282.19005)], leading to local index formulas. As an application to the classical family index theorem, it is shown that the characteristic numbers of the index bundle associated to a family of elliptic pseudodifferential operators are expressed in terms of the Wodzicki residue.