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On maximal ideals depending on some thin sets in $ M(G)$


Author: Enji Sato
Journal: Proc. Amer. Math. Soc. 87 (1983), 131-136
MSC: Primary 43A46; Secondary 43A10
DOI: https://doi.org/10.1090/S0002-9939-1983-0677248-4
MathSciNet review: 677248
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Abstract: Let $ M(G)$ be the convolution measure algebra on the LCA group $ G$ with dual $ \Gamma $. and $ \Delta $ the maximal ideal space of $ M(G)$. For $ E \subset G$ a compact set, let $ Gp(E)$ be the subgroup of $ G$ generated algebraically by $ E$. $ R(E)$ the measures which are carried by a countable union of translates of $ Gp(E)$. and $ {P_E}$ the natural projection from $ M(G)$ onto $ R(E)$. Also let $ {h_E}$ be the multiplicative linear functional $ \mu \mapsto ({P_E}\mu \hat )({\text{l}})$ on $ M(G)$. Then we prove that if $ G$ is an $ I$-group, and $ E$ an $ {H_1}$-set, we get $ {h_E} \in \bar \Gamma $ (i.e. the closure of $ \Gamma $ in $ \Delta $).


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  • [1] G. Brown, Idempotents in the closure of the characters, Bull. London Math. Soc. 4 (1972), 43-46. MR 0306815 (46:5937)
  • [2] C. F. Dunkl and D. E. Ramirez, Locally compact subgroup of the spectrum of the measure algebra, Semigroup Forum 3 (1971), 95-107. MR 0301450 (46:608)
  • [3] -, Bounded projections on Fourier-Stieltjes transform, Proc. Amer. Math. Soc. 31 (1972), 122-126. MR 0288520 (44:5718)
  • [4] I. Glicksberg and I. Wik, The range of Fourier-Stieltjes transforms of parts of measures, Conference on Harmonic Analysis, Maryland (1971), Lecture Notes in Math., vol. 266, Springer-Verlag. Berlin and New York, 1972. MR 0433145 (55:6124)
  • [5] C. C. Graham and O. C. McGehee, Essays in commutative harmonic analysis, Springer-Verlag, Berlin and New York, 1979. MR 550606 (81d:43001)
  • [6] T. W. Körner. Some results on Kronecker, Dirichlet and Helson sets, II, J. Analyse Math. 27 (1974), 260-388. MR 0487289 (58:6937)
  • [7] L. A. Lindahl and F. Poulsen (editors), Thin sets in harmonic analysis, Marcel Dekker, New York, 1972. MR 0393993 (52:14800)
  • [8] W. Rudin, Fourier analysis on groups, Interscience Tracts No 12. Interscience, New York, 1962. MR 0152834 (27:2808)
  • [9] S. Saeki, On the union of two Helson sets, J. Math. Soc. Japan 23 (1971), 636-648. MR 0293336 (45:2413)
  • [10] -, On strong Ditkin sets, Ark. Mat. 10 (1972), 1-7. MR 0310554 (46:9652)
  • [11] -, Symmetric maximal ideals in $ M(G)$, Pacific J. Math. 54 (1974), 229-243. MR 0370058 (51:6287)
  • [12] -, Bohr compactification and continuous measure, Proc. Amer. Math. Soc. 80 (1980), 244-246. MR 577752 (81i:43008)
  • [13] S. Saeki and E. Sato, Critical points and point derivations on $ M(G)$, Michigan Math. J. 25 (1978), 147-161. MR 501564 (80b:43002b)
  • [14] J. L. Taylor. Measure algebras. CBMS Regional Conf. Ser. in Math., no. 16, Amer. Math. Soc., Providence, R.I., 1973. MR 0427949 (55:979)
  • [15] N. Th. Varopoulos, Groups of continuous functions in harmonic analysis, Acta Math. 125 (1970), 109-152. MR 0282155 (43:7868)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0677248-4
Keywords: Measure algebra on LCA group, maximal ideal, Helson set
Article copyright: © Copyright 1983 American Mathematical Society

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