Dixmier traces and residues on weak operator ideals

TL;DR

This paper develops a comprehensive theory of modulated operators within principal ideals of compact operators, establishing trace formulas that connect Dixmier traces with Wodzicki residues, with applications in noncommutative geometry.

Contribution

It introduces a general framework for modulated operators, proves local and global trace formulas for Laplacian modulated operators, and links Dixmier traces to Wodzicki residues in noncommutative geometry.

Findings

Established Connes' trace formula for Laplacian modulated operators.

Connected Dixmier traces with Wodzicki residues in a broad setting.

Applied results to log-classical pseudo-differential operators and noncommutative geometry.

Abstract

We develop the theory of modulated operators in general principal ideals of compact operators. For Laplacian modulated operators we establish Connes' trace formula in its local Euclidean model and a global version thereof. It expresses Dixmier traces in terms of the vector-valued Wodzicki residue. We demonstrate the applicability of our main results in the context of log-classical pseudo-differential operators, studied by Lesch, and a class of operators naturally appearing in noncommutative geometry.