Helson Sets of Synthesis Are Ditkin Sets

Lau, Antony To-Ming; Ülger, Ali

doi:10.1090/proc/13887

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