When Kalton and Peck met Fourier

TL;DR

This paper explores the structure of short exact sequences of Banach modules over convolution algebras on compact abelian groups, introducing nonlinear $L_1$-centralizers and applying Fourier analysis to construct nontrivial sequences with specific properties.

Contribution

It introduces the concept of nonlinear $L_1$-centralizers in the context of Banach modules over convolution algebras and demonstrates their use in constructing nontrivial exact sequences.

Findings

Constructed nontrivial $L_1$-module sequences for various $p,q$ ranges.

Applied Fourier transform techniques to analyze module extensions.

Provided detailed examples for the circle and Cantor groups.

Abstract

The paper studies short exact sequences of Banach modules over the convolution algebra $L_1=L_1(G)$, where $G$ is a compact abelian group. The main tool is the notion of a nonlinear $L_1$-centralizer, which in combination with the Fourier transform, is used to produce sequences of $L_1$-modules $0\rightarrow L_q \rightarrow Z \rightarrow L_p \rightarrow 0$ that are nontrivial as long as the general theory allows it, namely for $p\in (1,\infty], q\in[1,\infty)$. Concrete examples are worked in detail for the circle group, with applications to the Hardy classes, and the Cantor group.