Measurable operators and the asymptotics of heat kernels and zeta functions

TL;DR

This paper explores the connection between measurable operators, as introduced by Alain Connes, and the asymptotic behavior of heat kernels and zeta functions, addressing longstanding open questions in noncommutative geometry.

Contribution

It provides new insights into the relationship between measurability and asymptotics of spectral functions, resolving questions that have persisted for over 15 years.

Findings

Established links between measurability and heat kernel asymptotics

Clarified the role of zeta function asymptotics in noncommutative geometry

Answered longstanding open questions about measurable operators

Abstract

In this note we answer some questions inspired by the introduction, by Alain Connes, of the notion of measurable operators using Dixmier traces. These questions concern the relationship of measurability to the asymptotics of $\zeta-$functions and heat kernels. The answers have remained elusive for some 15 years.