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Helson Sets of Synthesis Are Ditkin Sets

Authors: Antony To-Ming Lau and Ali Ülger
Journal: Proc. Amer. Math. Soc. 146 (2018), 2083-2090
MSC (2010): Primary 43A46, 43A45, 42A63; Secondary 43A20
Published electronically: December 11, 2017
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Abstract: Let $ G$ be a locally compact group and let $ A(G)$ be its Fourier algebra. A closed subset $ H$ of $ G$ is said to be a Helson set if the restriction homomorphism $ \phi :A(G)\rightarrow C_{0}(H)$, $ \phi (a)=a_{\vert H}$, is surjective. In this paper, under the hypothesis that $ G$ is amenable, we prove that every Helson subset $ H$ of $ G$ that is also a set of synthesis is a Ditkin set. This result is new even for $ G=\mathbb{R}$.

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Additional Information

Antony To-Ming Lau
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada T6G 2G1

Ali Ülger
Affiliation: Department of Mathematics, Bogazici University, 34342 Bebek/Istanbul, Turkey

Keywords: Helson set, set of synthesis, Ditkin set, amenable groups, Fourier algebra
Received by editor(s): June 27, 2016
Received by editor(s) in revised form: July 10, 2017
Published electronically: December 11, 2017
Additional Notes: The first author was supported by NSERC grant ZC912
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2017 American Mathematical Society

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