Extensions and renormalized traces

It has been shown by Nistor that given any extension of associative algebras over C, the connecting morphism in periodic cyclic homology is compatible, under the Chern-Connes character, with the index morphism in lower algebraic K-theory. The proof relies on the abstract properties of cyclic theory, essentially excision, which does not provide explicit formulas a priori. Avoiding the use of excision, we explain in this article how to get explicit formulas in a wide range of situations. The method is connected to the renormalization procedure introduced in our previous work on the bivariant Chern character for quasihomomorphisms, leading to “local” index formulas in the sense of non-commutative geometry. We illustrate these principles with the example of the classical family index theorem: we find that the characteristic numbers of the index bundle associated to a family of elliptic pseudodifferential operators are expressed in terms of the (fiberwise) Wodzicki residue.

💡 Research Summary

The paper revisits a fundamental compatibility result originally proved by V. Nistor: for any extension of associative complex algebras (0\to J\to A\to B\to0), the connecting morphism (\partial:HP_{}(B)\to HP_{+1}(J)) in periodic cyclic homology intertwines, via the Chern‑Connes character (\operatorname{ch}:K_{}\to HP_{}), with the index morphism (\operatorname{Ind}:K_{+1}(B)\to K_{}(J)) in lower algebraic K‑theory. Nistor’s proof relies heavily on the abstract excision property of cyclic homology, which guarantees the existence of (\partial) but does not provide concrete formulas. This lack of explicitness limits applications in non‑commutative geometry where one seeks “local” index formulas that can be evaluated on symbols of operators.

The authors propose a different route that avoids excision altogether. Building on their earlier work on the bivariant Chern character for quasihomomorphisms, they introduce a renormalization procedure that produces a regularized trace (\tau_{\mathrm{ren}}). The idea is to start from a quasihomomorphism ((\phi,\psi)) between two algebras, pass to the bivariant Chern character (\operatorname{Ch}^{\mathrm{biv}}(\phi,\psi)) in bivariant cyclic cohomology, and then eliminate the divergent pieces by inserting the Wodzicki residue – the unique trace on classical pseudodifferential operators of non‑integer order. This yields a concrete cochain representing the connecting morphism.

Concretely, after choosing a linear splitting (s:B\to A) of the projection (\pi:A\to B) and defining the difference map (h=a-s\pi(a)), the authors show that for a cyclic cycle (c\in C_{*}(B)) the connecting morphism can be written as \