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