Measures from Dixmier Traces and Zeta Functions

TL;DR

This paper investigates the relationship between noncommutative residues, Dixmier traces, and Lebesgue integrals on compact Riemannian manifolds, revealing limitations of the Dixmier trace in extending to integrable functions.

Contribution

It introduces symmetrised formulas for noncommutative residues and Dixmier traces, clarifying their applicability beyond square integrable functions and disproving previous claims of extension to all integrable functions.

Findings

Noncommutative residue equals Lebesgue integral for essentially bounded functions.

Symmetrised formulas extend to $L^{1+ ext{epsilon}}$-spaces but not to all integrable functions.

Dixmier trace diverges at integrable functions, unlike the noncommutative residue.

Abstract

For essentially bounded functions on a (closed) compact Riemannian manifold, the noncommutative residue and the Dixmier trace formulation of the noncommutative integral are shown to equate to a multiple of the Lebesgue integral. The identifications are shown to continue to, and be sharp at, square integrable functions. To do better than square integrable, symmetrised noncommutative residue and Dixmier trace formulas are introduced, for which the identifications are shown to continue to $L^{1+\epsilon}$-spaces, $\epsilon > 0$. However, a failure is shown for the Dixmier trace formulation at integrable functions. The (symmetrised) noncommutative residue and Dixmier trace formulas diverge at this point. It is shown the noncommutative residue remains finite and recovers the Lebesgue integral for any integrable function while the Dixmier trace expression can diverge. The results show the claim (in the monograph "Elements of Noncommutative Geometry", Birkhauser, 2001), that the identification on smooth functions obtained using Connes' Trace Theorem can be extended to any integrable function, is false.