Continuous measures and lacunarity on hypergroups

Vrem, Richard C.

doi:10.1090/S0002-9947-1982-0637708-2

Continuous measures and lacunarity on hypergroups
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Trans. Amer. Math. Soc. 269 (1982), 549-556 Request permission

Abstract:

The relationship between measures on a compact hypergroup $K$ whose Fourier-Stieltjes transforms vanish at infinity and the space ${M_c}(K)$ of continuous measures is studied. Examples are provided of measures $\mu$ with $\hat \mu$ vanishing at infinity and $\mu \in {M_c}(K)$. Sufficient conditions are given for $\hat \mu \in {c_0}(\hat K)$ to imply $\mu \in {M_c}(K)$. An investigation of Helson sets on compact abelian hypergroups is initiated and the study of Sidon sets on compact abelian hypergroups is continued. A class of compact abelian hypergroups is shown to have no infinite Helson sets and no infinite Sidon sets. This result generalizes results of D. L. Ragozin and D. Rider on central Sidon sets for compact connected Lie groups.

References

Spectral synthesis in products and quotients of hypergroups

A. H. Dooley, On lacunary sets for nonabelian groups, J. Austral. Math. Soc. Ser. A 25 (1978), no. 2, 167–176. MR 481935, DOI 10.1017/s144678870003874x
A. H. Dooley, Norms of characters and lacunarity for compact Lie groups, J. Functional Analysis 32 (1979), no. 2, 254–267. MR 534677, DOI 10.1016/0022-1236(79)90057-0
Charles F. Dunkl and Donald E. Ramirez, A family of countably compact $P_{\ast }$-hypergroups, Trans. Amer. Math. Soc. 202 (1975), 339–356. MR 380267, DOI 10.1090/S0002-9947-1975-0380267-9
Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, Die Grundlehren der mathematischen Wissenschaften, Band 152, Springer-Verlag, New York-Berlin, 1970. MR 0262773
Robert I. Jewett, Spaces with an abstract convolution of measures, Advances in Math. 18 (1975), no. 1, 1–101. MR 394034, DOI 10.1016/0001-8708(75)90002-X
J. F. Price, On local central lacunary sets for compact Lie groups, Monatsh. Math. 80 (1975), no. 3, 201–204. MR 390659, DOI 10.1007/BF01319915
David L. Ragozin, Central measures on compact simple Lie groups, J. Functional Analysis 10 (1972), 212–229. MR 0340965, DOI 10.1016/0022-1236(72)90050-x
Daniel Rider, Central lacunary sets, Monatsh. Math. 76 (1972), 328–338. MR 367559, DOI 10.1007/BF01297366
Kenneth A. Ross, Centers of hypergroups, Trans. Amer. Math. Soc. 243 (1978), 251–269. MR 493161, DOI 10.1090/S0002-9947-1978-0493161-2
Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0152834
Richard C. Vrem, Harmonic analysis on compact hypergroups, Pacific J. Math. 85 (1979), no. 1, 239–251. MR 571638
Richard C. Vrem, Lacunarity on compact hypergroups, Math. Z. 164 (1978), no. 2, 93–104. MR 517146, DOI 10.1007/BF01174816
Richard C. Vrem, Harmonic analysis on compact hypergroups, Pacific J. Math. 85 (1979), no. 1, 239–251. MR 571638