## Dual convolution for the affine group of the real line

#### Authors

Yemon Choi*, Mahya Ghandehari

* Corresponding author

Keywords: affine group, coefficient space, derivation, dual convolution, Fourier algebra, induced representation.

MSC 2020: 43A40, 47B90 (primary); 46J99 (secondary)

DOI: 10.1007/s11785-021-01100-y

#### Status

Published as Complex Anal. Oper. Theory 15 (2021), no. 4, article 76 (32 pages; open access)

N.B. In the bibliography, the items are listed in a slightly different order from what one would expect from this particular bibliographic style/format. However, the references within the document should be correct and consistent.

Preprint version available at arXiv 2009.05497

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

as a bilinear map $\left( L^p({\mathbb R}^\times) \hat\otimes L^q({\mathbb R}^\times) \right) \times \left( L^p({\mathbb R}^\times) \hat\otimes L^q({\mathbb R}^\times) \right) \to L^p({\mathbb R}^\times) \hat\otimes L^q({\mathbb R}^\times)$