The non-unital version of Connes' theorem on the Hochschild class of the Chern character

TL;DR

This paper provides a simplified proof of Connes' Hochschild class of the Chern character formula specifically for non-unital semifinite spectral triples, leveraging previous work on local index formulas and residues.

Contribution

It introduces a concise proof of the Hochschild class formula for non-unital spectral triples, building on refined local index formulas and singular trace representations.

Findings

Simplified proof of Connes' Hochschild class formula

Enhanced understanding of residues and singular traces

Application to non-unital semifinite spectral triples

Abstract

We offer a short proof of Connes' Hochschild class of the Chern character formula for non-unital semifinite spectral triples. The proof is simple due to its reliance on the authors' extensive work on a refined version of the local index formula, and the consequent understanding of the passage from generalised residues of zeta functions to representations in terms of singular traces.