Dual convolution for the affine group of the real line

Choi, Yemon and Ghandehari, Mahya (2021) Dual convolution for the affine group of the real line. Complex Analysis and Operator Theory, 15 (4). ISSN 1661-8262

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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 $p\in (1,2)\cup(2,\infty)$.

Item Type:
Journal Article
Journal or Publication Title:
Complex Analysis and Operator Theory
Uncontrolled Keywords:
/dk/atira/pure/subjectarea/asjc/2600/2604
Subjects:
ID Code:
152323
Deposited By:
Deposited On:
04 Mar 2021 12:25
Refereed?:
Yes
Published?:
Published
Last Modified:
20 Jun 2021 05:58