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Dual convolution for the affine group of the real line

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Dual convolution for the affine group of the real line. / Choi, Yemon; Ghandehari, Mahya.
In: Complex Analysis and Operator Theory, Vol. 15, No. 4, 76, 22.05.2021.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Choi, Y & Ghandehari, M 2021, 'Dual convolution for the affine group of the real line', Complex Analysis and Operator Theory, vol. 15, no. 4, 76. https://doi.org/10.1007/s11785-021-01100-y

APA

Choi, Y., & Ghandehari, M. (2021). Dual convolution for the affine group of the real line. Complex Analysis and Operator Theory, 15(4), Article 76. https://doi.org/10.1007/s11785-021-01100-y

Vancouver

Choi Y, Ghandehari M. Dual convolution for the affine group of the real line. Complex Analysis and Operator Theory. 2021 May 22;15(4):76. doi: 10.1007/s11785-021-01100-y

Author

Choi, Yemon ; Ghandehari, Mahya. / Dual convolution for the affine group of the real line. In: Complex Analysis and Operator Theory. 2021 ; Vol. 15, No. 4.

Bibtex

@article{4fe4173461764a2083f31326ba94c80e,
title = "Dual convolution for the affine group of the real line",
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 $p\in (1,2)\cup(2,\infty)$. ",
keywords = "affine group, coefficient space, derivation, dual convolution, Fourier algebra, induced representation",
author = "Yemon Choi and Mahya Ghandehari",
year = "2021",
month = may,
day = "22",
doi = "10.1007/s11785-021-01100-y",
language = "English",
volume = "15",
journal = "Complex Analysis and Operator Theory",
issn = "1661-8262",
publisher = "Springer",
number = "4",

}

RIS

TY - JOUR

T1 - Dual convolution for the affine group of the real line

AU - Choi, Yemon

AU - Ghandehari, Mahya

PY - 2021/5/22

Y1 - 2021/5/22

N2 - 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 $p\in (1,2)\cup(2,\infty)$.

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

KW - affine group

KW - coefficient space

KW - derivation

KW - dual convolution

KW - Fourier algebra

KW - induced representation

U2 - 10.1007/s11785-021-01100-y

DO - 10.1007/s11785-021-01100-y

M3 - Journal article

VL - 15

JO - Complex Analysis and Operator Theory

JF - Complex Analysis and Operator Theory

SN - 1661-8262

IS - 4

M1 - 76

ER -