Divided Differences in Noncommutative Geometry: Rearrangement Lemma, Functional Calculus and Expansional Formula

TL;DR

This paper generalizes the Rearrangement Lemma in noncommutative geometry, clarifies the multivariable functional calculus involved, and connects various formulas to divided differences, providing new insights and simplifications.

Contribution

It offers a simplified proof of the generalized Rearrangement Lemma, establishes a rigorous foundation for the multivariable functional calculus, and links expansion formulas to divided differences.

Findings

Generalization of the Rearrangement Lemma with a simple proof

Rigorous foundation for multivariable functional calculus in noncommutative geometry

Connection of expansion formulas to divided differences and Magnus expansion

Abstract

We state a generalization of the Connes-Tretkoff-Moscovici Rearrangement Lemma and give a surprisingly simple (almost trivial) proof of it. Secondly, we put on a firm ground the multivariable functional calculus used implicitly in the Rearrangement Lemma and elsewhere in the recent modular curvature paper by Connes and Moscovici. Furthermore, we show that the fantastic formulas connecting the one and two variable modular functions of loc. cit. are just examples of the plenty recursion formulas which can be derived from the calculus of divided differences. We show that the functions derived from the main integral occurring in the Rearrangement Lemma can be expressed in terms of divided differences of the Logarithm, generalizing the "modified Logarithm" of Connes-Tretkoff. Finally, we show that several expansion formulas related to the Magnus expansion have a conceptual explanation in terms of a multivariable functional calculus applied to divided differences.