## Helson Sets of Synthesis Are Ditkin Sets

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- by Antony To-Ming Lau and Ali Ülger PDF
- Proc. Amer. Math. Soc.
**146**(2018), 2083-2090 Request permission

## 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 $\phi :A(G)\rightarrow C_{0}(H)$, $\phi (a)=a_{|H}$, 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 $G=\mathbb {R}$.## References

- H. G. Dales,
*Banach algebras and automatic continuity*, London Mathematical Society Monographs. New Series, vol. 24, The Clarendon Press, Oxford University Press, New York, 2000. Oxford Science Publications. MR**1816726** - H. G. Dales and A. T.-M. Lau,
*The second duals of Beurling algebras*, Mem. Amer. Math. Soc.**177**(2005), no. 836, vi+191. MR**2155972**, DOI 10.1090/memo/0836 - Stephen William Drury,
*Sur les ensembles de Sidon*, C. R. Acad. Sci. Paris Sér. A-B**271**(1970), A162–A163 (French). MR**271647** - Pierre Eymard,
*L’algèbre de Fourier d’un groupe localement compact*, Bull. Soc. Math. France**92**(1964), 181–236 (French). MR**228628**, DOI 10.24033/bsmf.1607 - Henry Helson,
*Fourier transforms on perfect sets*, Studia Math.**14**(1954), 209–213 (1955). MR**68031**, DOI 10.4064/sm-14-2-209-213 - Eberhard Kaniuth and Anthony T. Lau,
*Spectral synthesis for $A(G)$ and subspaces of $VN(G)$*, Proc. Amer. Math. Soc.**129**(2001), no. 11, 3253–3263. MR**1845000**, DOI 10.1090/S0002-9939-01-05924-X - K. Kaniuth and A.T.-M. Lau,
*Fourier and Fourier-Stieltjes Algebras on Locally Compact Groups*, American Math. Society, Math. Surveys and Monographs 271 pages. - Eberhard Kaniuth and Ali Ülger,
*Weak spectral synthesis in commutative Banach algebras. III*, J. Funct. Anal.**268**(2015), no. 8, 2142–2170. MR**3318645**, DOI 10.1016/j.jfa.2015.01.004 - T. W. Körner,
*A Helson set of uniqueness but not of synthesis*, Colloq. Math.**62**(1991), no. 1, 67–71. MR**1114620**, DOI 10.4064/cm-62-1-67-71 - Horst Leptin,
*Sur l’algèbre de Fourier d’un groupe localement compact*, C. R. Acad. Sci. Paris Sér. A-B**266**(1968), A1180–A1182 (French). MR**239002** - Paul Malliavin,
*Impossibilité de la synthèse spectrale sur les groupes abéliens non compacts*, Séminaire P. Lelong, 1958/59, exp. 17, Faculté des Sciences de Paris, 1959, pp. 8 (French). MR**0107126** - Walter Rudin,
*Fourier analysis on groups*, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1990. Reprint of the 1962 original; A Wiley-Interscience Publication. MR**1038803**, DOI 10.1002/9781118165621 - Sadahiro Saeki,
*Spectral synthesis for the Kronecker sets*, J. Math. Soc. Japan**21**(1969), 549–563. MR**254525**, DOI 10.2969/jmsj/02140549 - Sadahiro Saeki,
*A characterization of $\textrm {SH}$-sets*, Proc. Amer. Math. Soc.**30**(1971), 497–503. MR**283500**, DOI 10.1090/S0002-9939-1971-0283500-9 - Sadahiro Saeki,
*Extremally disconnected sets in groups*, Proc. Amer. Math. Soc.**52**(1975), 317–318. MR**372541**, DOI 10.1090/S0002-9939-1975-0372541-2 - N. Th. Varopoulos,
*Groups of continuous functions in harmonic analysis*, Acta Math.**125**(1970), 109–154. MR**282155**, DOI 10.1007/BF02392332

## Additional Information

**Antony To-Ming Lau**- Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada T6G 2G1
- MR Author ID: 110640
- Email: antonyt@ualberta.ca
**Ali Ülger**- Affiliation: Department of Mathematics, Bogazici University, 34342 Bebek/Istanbul, Turkey
- Email: aulger@ku.edu.tr
- 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
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**146**(2018), 2083-2090 - MSC (2010): Primary 43A46, 43A45, 42A63; Secondary 43A20
- DOI: https://doi.org/10.1090/proc/13887
- MathSciNet review: 3767359