Universal measurability and the Hochschild class of the Chern character

TL;DR

This paper investigates measurability of singular traces and applies these results to clarify the Hochschild class of the Chern character in spectral triples, showing independence from the choice of trace and providing eigenvalue asymptotics.

Contribution

It introduces a characterization of universal measurability for operators in Dixmier ideals and applies this to improve proofs regarding the Hochschild class of the Chern character.

Findings

Measurability results are established for singular traces.

The Hochschild class identification is shown to be trace-independent.

Eigenvalue asymptotics for operators associated with spectral triples are derived.

Abstract

We study notions of measurability for singular traces, and characterise universal measurability for operators in Dixmier ideals. This measurability result is then applied to improve on the various proofs of Connes' identification of the Hochschild class of the Chern character of Dixmier summable spectral triples. The measurability results show that the identification of the Hochschild class is independent of the choice of singular trace. As a corollary we obtain strong information on the asymptotics of the eigenvalues of operators naturally associated to spectral triples (A,H,D) and Hochschild cycles for A.