Dual convolution for the affine group of the real line

TL;DR

This paper investigates the dual convolution product in the Fourier algebra of the affine group of the real line, providing an intrinsic description, explicit derivations, and extending the analysis to trace-class operators on L^p spaces.

Contribution

It offers an intrinsic characterization of the dual convolution, constructs explicit derivations, and extends the Banach algebra structure analysis to L^p spaces beyond L^2.

Findings

Intrinsic description of the dual convolution product

Explicit derivations on the Banach algebras

Extension to trace-class operators on L^p spaces

Abstract

The Fourier algebra of the affine group of the real line has a natural identification, as a Banach space, with the space of trace-class operators on $L^2({\mathbb R}^\times, dt/ |t|)$. In this paper we study the "dual convolution product" of trace-class operators that corresponds to pointwise product in the Fourier algebra. Answering a question raised in work of Eymard and Terp, we provide an intrinsic description of this operation which does not rely on the identification with the Fourier algebra, and obtain a similar result for the connected component of this affine group. In both cases we construct explicit derivations on the corresponding Banach algebras, verifying the derivation identity directly without requiring the inverse Fourier transform. We also initiate the study of the analogous Banach algebra structure for trace-class operators on $L^p({\mathbb R}^\times, dt/ |t|)$ for $p\in (1,2)\cup(2,\infty)$.